examples/notebooks/spatial_methods/finite-volumes.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finite Volume Discretisation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we explain the discretisation process that converts an expression tree, representing a model, to a linear algebra tree that can be evaluated by the solvers. \n",
    "\n",
    "We use Finite Volumes as an example of a spatial method, since it is the default spatial method for most PyBaMM models. This is a good spatial method for battery problems as it is conservative: for lithium-ion battery models, we can be sure that the total amount of lithium in the system is constant. For more details on the Finite Volume method, see [Randall Leveque's book](https://books.google.co.uk/books/about/Finite_Volume_Methods_for_Hyperbolic_Pro.html?id=QazcnD7GUoUC&printsec=frontcover&source=kp_read_button&redir_esc=y#v=onepage&q&f=false).\n",
    "\n",
    "This notebook is structured as follows:\n",
    "\n",
    "1. **Setting up a discretisation**. Overview of the parameters that are passed to the discretisation\n",
    "2. **Discretisations and spatial methods**. Operations that are common to most spatial methods:\n",
    "    - Discretising a spatial variable (e.g. $x$)\n",
    "    - Discretising a variable (e.g. concentration)\n",
    "3. **Example: Finite Volume operators**. Finite Volume implementation of some useful operators: \n",
    "    - Gradient operator\n",
    "    - Divergence operator\n",
    "    - Integral operator\n",
    "4. **Example: Discretising a simple model**. Setting up and solving a simple model, using Finite Volumes as the spatial method\n",
    "\n",
    "To find out how to implement a new spatial method, see the [tutorial](https://pybamm.readthedocs.io/en/latest/tutorials/add-spatial-method.html) in the API docs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setting up a Discretisation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We first import `pybamm` and some useful other modules, and change our working directory to the root of the `PyBaMM` folder:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[33mWARNING: You are using pip version 20.2.4; however, version 21.0.1 is available.\n",
      "You should consider upgrading via the '/Users/vsulzer/Documents/Energy_storage/PyBaMM/.tox/dev/bin/python -m pip install --upgrade pip' command.\u001b[0m\n",
      "Note: you may need to restart the kernel to use updated packages.\n"
     ]
    }
   ],
   "source": [
    "%pip install pybamm -q    # install PyBaMM if it is not installed\n",
    "import pybamm\n",
    "import numpy as np\n",
    "import os\n",
    "import matplotlib.pyplot as plt\n",
    "from pprint import pprint\n",
    "os.chdir(pybamm.__path__[0]+'/..')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To set up a discretisation, we must create a geometry, mesh this geometry, and then create the discretisation with the appropriate spatial method(s). The easiest way to create a geometry is to the inbuilt battery geometry:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "parameter_values = pybamm.ParameterValues(\n",
    "    values={\n",
    "        \"Negative electrode thickness [m]\": 0.3,\n",
    "        \"Separator thickness [m]\": 0.2,\n",
    "        \"Positive electrode thickness [m]\": 0.3,\n",
    "    }\n",
    ")\n",
    "\n",
    "geometry = pybamm.battery_geometry()\n",
    "parameter_values.process_geometry(geometry)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then use this geometry to create a mesh, which for this example consists of uniform 1D submeshes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "submesh_types = {\n",
    "    \"negative electrode\": pybamm.Uniform1DSubMesh,\n",
    "    \"separator\": pybamm.Uniform1DSubMesh,\n",
    "    \"positive electrode\": pybamm.Uniform1DSubMesh,\n",
    "    \"negative particle\": pybamm.Uniform1DSubMesh,\n",
    "    \"positive particle\": pybamm.Uniform1DSubMesh,\n",
    "    \"current collector\": pybamm.SubMesh0D,\n",
    "}\n",
    "\n",
    "var = pybamm.standard_spatial_vars\n",
    "var_pts = {var.x_n: 15, var.x_s: 10, var.x_p: 15, var.r_n: 10, var.r_p: 10}\n",
    "mesh = pybamm.Mesh(geometry, submesh_types, var_pts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can use the mesh to create a discretisation, using Finite Volumes as the spatial method for this example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "spatial_methods = {\n",
    "    \"macroscale\": pybamm.FiniteVolume(),\n",
    "    \"negative particle\": pybamm.FiniteVolume(),\n",
    "    \"positive particle\": pybamm.FiniteVolume(),\n",
    "}\n",
    "disc = pybamm.Discretisation(mesh, spatial_methods)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discretisations and Spatial Methods"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Spatial Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Spatial variables, such as $x$ and $r$, are converted to `pybamm.Vector` nodes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_disc is a <class 'pybamm.expression_tree.vector.Vector'>\n",
      "r_disc is a <class 'pybamm.expression_tree.vector.Vector'>\n"
     ]
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 936x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set up \n",
    "macroscale = [\"negative electrode\", \"separator\", \"positive electrode\"]\n",
    "x_var = pybamm.SpatialVariable(\"x\", domain=macroscale)\n",
    "r_var = pybamm.SpatialVariable(\"r\", domain=[\"negative particle\"])\n",
    "\n",
    "# Discretise\n",
    "x_disc = disc.process_symbol(x_var)\n",
    "r_disc = disc.process_symbol(r_var)\n",
    "print(\"x_disc is a {}\".format(type(x_disc)))\n",
    "print(\"r_disc is a {}\".format(type(r_disc)))\n",
    "\n",
    "# Evaluate\n",
    "x = x_disc.evaluate()\n",
    "r = r_disc.evaluate()\n",
    "\n",
    "f, (ax1, ax2) = plt.subplots(1, 2, figsize=(13,4))\n",
    "\n",
    "ax1.plot(x, \"*\")\n",
    "ax1.set_xlabel(\"index\")\n",
    "ax1.set_ylabel(r\"$x$\")\n",
    "\n",
    "ax2.plot(r, \"*\")\n",
    "ax2.set_xlabel(\"index\")\n",
    "ax2.set_ylabel(r\"$r$\")\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define `y_macroscale`, `y_microscale` and `y_scalar` for evaluation and visualisation of results below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_macroscale = x ** 3 / 3\n",
    "y_microscale = np.cos(r)\n",
    "y_scalar = np.array([[5]])\n",
    "\n",
    "y = np.concatenate([y_macroscale, y_microscale, y_scalar])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we will work with three variables `u`, `v`, `w`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = pybamm.Variable(\"u\", domain=macroscale) # u is a variable in the macroscale (e.g. electrolyte potential)\n",
    "v = pybamm.Variable(\"v\", domain=[\"negative particle\"]) # v is a variable in the negative particle (e.g. particle concentration)\n",
    "w = pybamm.Variable(\"w\") # w is a variable without a domain (e.g. time, average concentration)\n",
    "\n",
    "variables = [u,v,w]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before discretising, trying to evaluate the variables raises a `NotImplementedError`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "method self.evaluate() not implemented\n",
      "               for symbol u of type <class 'pybamm.expression_tree.variable.Variable'>\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    u.evaluate()\n",
    "except NotImplementedError as e: \n",
    "    print(e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For any spatial method, a `pybamm.Variable` gets converted to a `pybamm.StateVector` which, when evaluated, takes the appropriate slice of the input vector `y`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Discretised u is the StateVector y[0:40]\n",
      "Discretised v is the StateVector y[40:50]\n",
      "Discretised w is the StateVector y[50:51]\n"
     ]
    }
   ],
   "source": [
    "# Pass the list of variables to the discretisation to calculate the slices to be used (order matters here!)\n",
    "disc.set_variable_slices(variables)\n",
    "\n",
    "# Discretise the variables\n",
    "u_disc = disc.process_symbol(u)\n",
    "v_disc = disc.process_symbol(v)\n",
    "w_disc = disc.process_symbol(w)\n",
    "\n",
    "# Print the outcome \n",
    "print(\"Discretised u is the StateVector {}\".format(u_disc))\n",
    "print(\"Discretised v is the StateVector {}\".format(v_disc))\n",
    "print(\"Discretised w is the StateVector {}\".format(w_disc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the variables have been passed to `disc` in the order `[u,v,w]`, they each read the appropriate part of `y` when evaluated:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 936x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_fine = np.linspace(x[0], x[-1], 1000)\n",
    "r_fine = np.linspace(r[0], r[-1], 1000)\n",
    "\n",
    "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13,4))\n",
    "ax1.plot(x_fine, x_fine**3/3, x, u_disc.evaluate(y=y), \"o\")\n",
    "ax1.set_xlabel(\"x\")\n",
    "ax1.legend([\"x^3/3\", \"u\"], loc=\"best\")\n",
    "\n",
    "ax2.plot(r_fine, np.cos(r_fine), r, v_disc.evaluate(y=y), \"o\")\n",
    "ax2.set_xlabel(\"r\")\n",
    "ax2.legend([\"cos(r)\", \"v\"], loc=\"best\")\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w = [[5.]]\n"
     ]
    }
   ],
   "source": [
    "print(\"w = {}\".format(w_disc.evaluate(y=y)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finite Volume Operators"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gradient operator\n",
    "\n",
    "The gradient operator is converted to a Matrix-StateVector multiplication. In 1D, the gradient operator is equivalent to $\\partial/\\partial x$ on the macroscale and $\\partial/\\partial r$ on the microscale. In Finite Volumes, we take the gradient of an object on nodes (shape (n,)), which returns an object on the edges (shape (n-1,)). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@\n",
      "├── Sparse Matrix (39, 40)\n",
      "└── y[0:40]\n"
     ]
    }
   ],
   "source": [
    "grad_u = pybamm.grad(u)\n",
    "grad_u_disc = disc.process_symbol(grad_u)\n",
    "grad_u_disc.render()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Matrix in `grad_u_disc` is the standard `[-1,1]` sparse matrix, divided by the step sizes `dx`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gradient matrix is:\n",
      "\n",
      "1/dx *\n",
      "[[-1.  1.  0. ...  0.  0.  0.]\n",
      " [ 0. -1.  1. ...  0.  0.  0.]\n",
      " [ 0.  0. -1. ...  0.  0.  0.]\n",
      " ...\n",
      " [ 0.  0.  0. ...  1.  0.  0.]\n",
      " [ 0.  0.  0. ... -1.  1.  0.]\n",
      " [ 0.  0.  0. ...  0. -1.  1.]]\n"
     ]
    }
   ],
   "source": [
    "macro_mesh = mesh.combine_submeshes(*macroscale)\n",
    "print(\"gradient matrix is:\\n\")\n",
    "print(\"1/dx *\\n{}\".format(macro_mesh.d_nodes[:,np.newaxis] * grad_u_disc.children[0].entries.toarray()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When evaluated with `y_macroscale=x**3/3`, `grad_u_disc` is equal to `x**2` as expected:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_edge = macro_mesh.edges[1:-1] # note that grad_u_disc is evaluated on the node edges\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x_fine, x_fine**2, x_edge, grad_u_disc.evaluate(y=y), \"o\")\n",
    "ax.set_xlabel(\"x\")\n",
    "legend = ax.legend([\"x^2\", \"grad(u).evaluate(y=x**3/3)\"], loc=\"best\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similary, we can create, discretise and evaluate the gradient of `v`, which is a variable in the negative particles. Note that the syntax for doing this is identical: we do not need to explicitly specify that we want the gradient in `r`, since this is inferred from the `domain` of `v`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['negative particle']"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.domain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "grad(v) tree is:\n",
      "\n",
      "@\n",
      "├── Sparse Matrix (9, 10)\n",
      "└── y[40:50]\n",
      "\n",
      " gradient matrix is:\n",
      "\n",
      "1/dr *\n",
      "[[-1.  1.  0.  0.  0.  0.  0.  0.  0.  0.]\n",
      " [ 0. -1.  1.  0.  0.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0. -1.  1.  0.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  0. -1.  1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0. -1.  1.  0.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0. -1.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  0. -1.  1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  0.  0. -1.  1.  0.]\n",
      " [ 0.  0.  0.  0.  0.  0.  0.  0. -1.  1.]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "grad_v = pybamm.grad(v)\n",
    "grad_v_disc = disc.process_symbol(grad_v)\n",
    "print(\"grad(v) tree is:\\n\")\n",
    "grad_v_disc.render()\n",
    "\n",
    "micro_mesh = mesh[\"negative particle\"]\n",
    "print(\"\\n gradient matrix is:\\n\")\n",
    "print(\"1/dr *\\n{}\".format(micro_mesh.d_nodes[:,np.newaxis] * grad_v_disc.children[0].entries.toarray()))\n",
    "\n",
    "r_edge = micro_mesh.edges[1:-1] # note that grad_u_disc is evaluated on the node edges\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(r_fine, -np.sin(r_fine), r_edge, grad_v_disc.evaluate(y=y), \"o\")\n",
    "ax.set_xlabel(\"x\")\n",
    "legend = ax.legend([\"-sin(r)\", \"grad(v).evaluate(y=cos(r))\"], loc=\"best\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Boundary conditions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the discretisation is provided with boundary conditions, appropriate ghost nodes are concatenated onto the variable, and a larger gradient matrix is used. The ghost nodes are chosen based on the value of the first/last node in the variable and the boundary condition.\n",
    "For a Dirichlet boundary condition $u=a$ on the left-hand boundary, we set the value of the left ghost node to be equal to\n",
    "$$2*a-u[0],$$\n",
    "where $u[0]$ is the value of $u$ in the left-most cell in the domain. Similarly, for a Dirichlet condition $u=b$ on the right-hand boundary, we set the right ghost node to be\n",
    "$$2*b-u[-1].$$\n",
    "Note also that the size of the gradient matrix is now (41,42) instead of (39,40), to account for the presence of boundary conditions in the State Vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The gradient object is:\n",
      "+\n",
      "├── @\n",
      "│   ├── Sparse Matrix (41, 40)\n",
      "│   └── y[0:40]\n",
      "└── Column vector of length 41\n",
      "The value of u on the left-hand boundary is [-40.]\n",
      "The value of u on the right-hand boundary is [80.]\n"
     ]
    }
   ],
   "source": [
    "disc.bcs = {u: {\"left\": (pybamm.Scalar(1), \"Dirichlet\"), \"right\": (pybamm.Scalar(2), \"Dirichlet\")}}\n",
    "grad_u_disc = disc.process_symbol(grad_u)\n",
    "print(\"The gradient object is:\")\n",
    "(grad_u_disc.render())\n",
    "u_eval = grad_u_disc.children[1].evaluate(y=y)\n",
    "print(\"The value of u on the left-hand boundary is {}\".format((u_eval[0] + u_eval[1]) / 2))\n",
    "print(\"The value of u on the right-hand boundary is {}\".format((u_eval[-2] + u_eval[-1]) / 2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a Neumann boundary condition $\\partial u/\\partial x=c$ on the left-hand boundary, we set the value of the left ghost node to be\n",
    "$$u[0] - c * dx,$$\n",
    "where $dx$ is the step size at the left-hand boundary. For a Neumann boundary condition $\\partial u/\\partial x=d$ on the right-hand boundary, we set the value of the right ghost node to be\n",
    "$$u[-1] + d * dx.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The gradient object is:\n",
      "+\n",
      "├── @\n",
      "│   ├── Sparse Matrix (41, 40)\n",
      "│   └── y[0:40]\n",
      "└── Column vector of length 41\n",
      "The gradient on the left-hand boundary is [3.]\n",
      "The gradient of u on the right-hand boundary is [4.]\n"
     ]
    }
   ],
   "source": [
    "disc.bcs = {u: {\"left\": (pybamm.Scalar(3), \"Neumann\"), \"right\": (pybamm.Scalar(4), \"Neumann\")}}\n",
    "grad_u_disc = disc.process_symbol(grad_u)\n",
    "print(\"The gradient object is:\")\n",
    "(grad_u_disc.render())\n",
    "grad_u_eval = grad_u_disc.evaluate(y=y)\n",
    "print(\"The gradient on the left-hand boundary is {}\".format(grad_u_eval[0]))\n",
    "print(\"The gradient of u on the right-hand boundary is {}\".format(grad_u_eval[-1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can mix the types of the boundary conditions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The gradient object is:\n",
      "+\n",
      "├── +\n",
      "│   ├── @\n",
      "│   │   ├── Sparse Matrix (41, 40)\n",
      "│   │   └── y[0:40]\n",
      "│   └── Column vector of length 41\n",
      "└── Column vector of length 41\n",
      "The value of u on the left-hand boundary is [0.]\n",
      "The gradient on the right-hand boundary is [6.]\n"
     ]
    }
   ],
   "source": [
    "disc.bcs = {u: {\"left\": (pybamm.Scalar(5), \"Dirichlet\"), \"right\": (pybamm.Scalar(6), \"Neumann\")}}\n",
    "grad_u_disc = disc.process_symbol(grad_u)\n",
    "print(\"The gradient object is:\")\n",
    "(grad_u_disc.render())\n",
    "grad_u_eval = grad_u_disc.evaluate(y=y)\n",
    "u_eval = grad_u_disc.children[1].evaluate(y=y)\n",
    "print(\"The value of u on the left-hand boundary is {}\".format((u_eval[0] + u_eval[1])/2))\n",
    "print(\"The gradient on the right-hand boundary is {}\".format(grad_u_eval[-1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Robin boundary conditions can be implemented by specifying a Neumann condition where the flux depends on the variable."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Divergence operator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before computing the Divergence operator, we set up Neumann boundary conditions. The behaviour with Dirichlet boundary conditions is very similar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "disc.bcs = {u: {\"left\": (pybamm.Scalar(-1), \"Neumann\"), \"right\": (pybamm.Scalar(1), \"Neumann\")}}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can process `div(grad(u))`, converting it to a Matrix-Vector multiplication, plus a vector for the boundary conditions. Since we have Neumann boundary conditions, the divergence of an object of size (n+1,) has size (n,), and so div(grad) of an object of size (n,) has size (n,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+\n",
      "├── @\n",
      "│   ├── Sparse Matrix (40, 40)\n",
      "│   └── y[0:40]\n",
      "└── Column vector of length 40\n"
     ]
    }
   ],
   "source": [
    "div_grad_u = pybamm.div(grad_u)\n",
    "div_grad_u_disc = disc.process_symbol(div_grad_u)\n",
    "div_grad_u_disc.render()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The div(grad) matrix is automatically simplified to the well-known `[1,-2,1]` matrix (divided by the square of the distance between the edges), except in the first and last rows for boundary conditions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "div(grad) matrix is:\n",
      "\n",
      "1/dx^2 * \n",
      "[[-1.  1.  0. ...  0.  0.  0.]\n",
      " [ 1. -2.  1. ...  0.  0.  0.]\n",
      " [ 0.  1. -2. ...  0.  0.  0.]\n",
      " ...\n",
      " [ 0.  0.  0. ... -2.  1.  0.]\n",
      " [ 0.  0.  0. ...  1. -2.  1.]\n",
      " [ 0.  0.  0. ...  0.  1. -1.]]\n"
     ]
    }
   ],
   "source": [
    "print(\"div(grad) matrix is:\\n\")\n",
    "print(\"1/dx^2 * \\n{}\".format(\n",
    "    macro_mesh.d_edges[:,np.newaxis]**2 * div_grad_u_disc.left.left.entries.toarray()\n",
    "))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integral operator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can define an integral operator, which integrates the variable across the domain specified by the integration variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "int(u) = [[0.08330729]] is approximately equal to 1/12, 0.08333333333333333\n",
      "int(v/r^2) = [[10.57864347]] is approximately equal to 4 * pi * sin(1), 10.574236256325824\n"
     ]
    }
   ],
   "source": [
    "int_u = pybamm.Integral(u, x_var)\n",
    "int_u_disc = disc.process_symbol(int_u)\n",
    "print(\"int(u) = {} is approximately equal to 1/12, {}\".format(int_u_disc.evaluate(y=y), 1/12))\n",
    "\n",
    "# We divide v by r to evaluate the integral more easily\n",
    "int_v_over_r2 = pybamm.Integral(v/r_var**2, r_var)\n",
    "int_v_over_r2_disc = disc.process_symbol(int_v_over_r2)\n",
    "print(\"int(v/r^2) = {} is approximately equal to 4 * pi * sin(1), {}\".format(\n",
    "    int_v_over_r2_disc.evaluate(y=y), 4 * np.pi * np.sin(1))\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The integral operators are also Matrix-Vector multiplications"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "int(u):\n",
      "\n",
      "@\n",
      "├── Sparse Matrix (1, 40)\n",
      "└── y[0:40]\n",
      "\n",
      "int(v):\n",
      "\n",
      "@\n",
      "├── Sparse Matrix (1, 10)\n",
      "└── y[40:50]\n"
     ]
    }
   ],
   "source": [
    "print(\"int(u):\\n\")\n",
    "int_u_disc.render()\n",
    "print(\"\\nint(v):\\n\")\n",
    "int_v_over_r2_disc.render()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,\n",
       "         1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,\n",
       "         1., 1., 1., 1., 1., 1., 1., 1.]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int_u_disc.children[0].evaluate() / macro_mesh.d_edges"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[12.56637061, 12.56637061, 12.56637061, 12.56637061, 12.56637061,\n",
       "         12.56637061, 12.56637061, 12.56637061, 12.56637061, 12.56637061]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int_v_over_r2_disc.children[0].evaluate() / micro_mesh.d_edges"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discretising a model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now discretise a whole model. We create, and discretise, a simple model for the concentration in the electrolyte and the concentration in the particles, and discretise it with a single command:\n",
    "```\n",
    "disc.process_model(model)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = pybamm.BaseModel()\n",
    "\n",
    "c_e = pybamm.Variable(\"electrolyte concentration\", domain=macroscale)\n",
    "N_e = pybamm.grad(c_e)\n",
    "c_s = pybamm.Variable(\"particle concentration\", domain=[\"negative particle\"])\n",
    "N_s = pybamm.grad(c_s)\n",
    "model.rhs = {c_e: pybamm.div(N_e) - 5, c_s: pybamm.div(N_s)}\n",
    "model.boundary_conditions = {\n",
    "    c_e: {\"left\": (np.cos(0), \"Neumann\"), \"right\": (np.cos(10), \"Neumann\")},\n",
    "    c_s: {\"left\": (0, \"Neumann\"), \"right\": (-1, \"Neumann\")},\n",
    "}\n",
    "model.initial_conditions = {c_e: 1 + 0.1 * pybamm.sin(10*x_var), c_s: 1}\n",
    "\n",
    "# Create a new discretisation and process model\n",
    "disc2 = pybamm.Discretisation(mesh, spatial_methods)\n",
    "disc2.process_model(model);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The initial conditions are discretised to vectors, and an array of concatenated initial conditions is created."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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UBdwYULG/hhnz36ZCczRFYpaKELXEK/+eUncYDHrW7Yp6+fd+PTI5e2guz7+7g//8zHACtU4knj3/7g5O7nMSIwf0iPp7X3bmQH7y0gc8/+4Ovn7+KVF//wRWDgRP6M3z2s4BPm1mtwLdgS5mdtA5N7P5DpxzDwEPARQVFbnIhyySvG644QZuuOGGE67zl7/8JeT9zZo1i1mzZnU2LAmjeUs2smJrFfNe2cjcK87wOxwRaUGbPaBm9oiZVZjZ+60sNzObZ2abzKzEzMZ57VPMbE3Qo8bMpnvLHjWzLUHLxoTzQ3VahMq/t9dlZw5k8+6PWbtjf1TfV8KsZCENPx3Fn8ov4q9H/h1776moh5CfexJjBufw3Br1qIfZc8D13nHwbKDaObfTOXetc26Ic66AwDDcx1pKPkVEoiEZzuVG3LOYgpkvsGBZKc7BgmWlFMx8gRH3LPYzLOkk9WgnplCG4D5KB6pAOudec86Ncc6NAS4gcIuC4GIc/9203Dm3pv2hR1AEyr93xCWj+5OWYhqGG8+83vTUA2WkGPSo/SjQm16yMOqhXD5mIOt27mdTxYGov3e8MrPHgbeBEWZWZmY3m9nXzezr3iqLgM3AJuB3BIbeikgcuO+++xgzZswxj/vuu++49V588cXj1rviiit8iLhTHiXBz+WW3jWFaWMGkpkeOLXNTE/h8jEDWXr3FD/Dkk4K7tGWxNHmEFzn3BtmVnCCVY5WgQTeMbMcMxvgnNsZtM4XgMXOuUOdCzdKsvOguoWiA50o/94RPU/qwnmn9uH5d3dw98WnkZKiYbhx50S96VG+xc6lhQP4/t/W8dyaHXzrsyOi+t7xyjl3dRvLHfDNNtZ5lMDJn0jSc87FzJSSUIfPXnTRRVx00UUdfp/AYcJfyXAu17dHJlkZadTWN5KRlkJtfSNZGWn0zcr0OzTpgBH3LKa2vvHo6wXLSlmwrJSMtBQ2zL3Ex8gkHMJRhKjFKpDN1rkKeLxZ233eMI8HzCyjtZ37UqBj6mxqm4cUpfLvzV125gB2VNewevu+qL+3hEGM9KYD9M3K5M5+a7j27Uvh3hx4YLQvPbEikpwyMzPZs2dPTCRk0eKcY8+ePWRmxnwSFLFzuWiex+0+WMu1E/N55tZJXDsxn8qDtRF9P4kcv3q0NeQ3OiJehMi7J94ZwItBzd8BPgK6ECi+cTfQ4gRLPwp07Bs2nXvr1vC9bv9H9pFdgZ7PqbOj3mMFcMFp/UhLMV5et4uz8ntG/f2lk2KkNx2AkoV8Y/880px3UK3eHvXiWiKSvPLy8igrKyPZqj1nZmaSl+fDMT+MOnMuF83zuPnXFR19Pnf66Ei+lUSYXz3aKmIVHeFIQFurAtlkBvCMc66uqSFoSEetmf2eGLtX3qv/quCv9ZO48bq7GDM4x9dYsrumc84puby09iPuvnhEzAxdkhBNnU3NX24jk6CrsD71prNkDmmNza7o+TQcWESST3p6OkOHDvU7DGlZwp3LSfxr6tG+ZsIQ/ry8lMoI9kpqyG90hWMIbotVIIOWX02zIRvelTQskE1NB1qsyuaXl9ftol+PDAoHZfsdCgC35BTz2P6b4Xs9NWwyzlSdMp2ZdTdTndGfcN1MvcNiaDiwiIjElIQ7l4sWDdmMnPnXFTF3+mhGDuzB3Omjj+nhDjcVsYquNntAvSqQk4HeZlYGfBdIB3DO/ZZAFcjPEagCeQi4KWjbAgJX1P7RbLd/MrM+gAFrgK8TI2rqGvjHB5VcOW5QbBT9KVnIpPXfJyXFK2SjYZNx5dV/VfDXhk9x05fv5kyfe9NjajiwiIhETbKdy0WThmwmBhWxiq5QquB2uAqkc24rx09ixzl3QYjxRd2bm3Zz6EgDnx3Z3+9QApbMIaU+NqqoSvu94vWmnxELvelTZwcuXgRX5fVrOLCIiERNsp3LRYOGbCaeaA75TXYRL0IUb15au4usjDTOPjnX71ACNGwybtXUNfDGxkquGBsjveneBQu3ZA6uuoyq1D70vuw+XcgQERFpp6V3TWHuovW8tPYjauoayUxP4aJR/Zl16el+hyYdpCJW0ROOOaAJo6HR8cr6XUw+rS9d0mLkV9Pa8EgNm4x5b30Y6E2/cGQ/v0P5ROEM7M73uX/8W5xbM4+PR1zpd0QiIiJxR0M2RTouRrKs2PBeeTV7Pj7CZ07v63con5g6OzBMMpiGTcaFl9ft4qQuqZxzSoz0pgeZPKIvRxoaeevDPX6HIiIiEpd031GRjtEQ3CCvb6jADD49vI/foXyiaXikN2xyh8ul76X3ka5hkzHNOcer/6rgvFP7kJGW6nc4xykq6MlJXVJ5fUNFbPXQioiIxAkN2RTpGPWABvnHB5WcmZdDr5O6+B3KsQpnwJ3v8+qXNjCpdh7LTvqM3xFJGzbsOsCu/bVMHhFDFzOCZKSlcu6w3ry+oZJA7QkRERERkchTAurZ+/ER3t2+j/NPjc2EAeCcU3LpkprC6xsq/A5F2vDGB5UAnBfD36fJI/pQvu8wmyoO+h2KiIiIiCQJJaCepZt20+jg/BjtsQLo1iWNCUN78bqX3Ejs+scHlZzarzsDsru2vbJPJo8IzHV+fYO+TyIiIiISHUpAPf/YUElOt3TOzMvxO5QTmjyiD5sqDlK295DfoUgrDh2pZ8WWvTHdmw4wKKcrp/brzmvqURcRERGRKFECWrIQ98BofrzufF5NuY3U95/yO6ITappT+A/1gsasZZurONLQGNPDb5tMHtGXFVurOFhb73coIiIiIpIEkjsBLVkIz9+OVW8nBUev+l3w/O2B9hh1Sp/uDMrpqmGTMewfH1SSmZ7C+IJefofSpsmn9qGuwfHWpt1+hyIiItIuFftrmDH/bSoO1Pgdioi0Q3InoEvmQN3hY9vqDgfaY5SZMXlEH97atJsj9Y1+hyMteOODSs4+OZfM9Ni7/UpzRQW96Jqeyj+VgIqISJyZt2QjK7ZWMe+VjX6HIiLtkNwJaHVZ+9pjxKeH9+bjIw28W7bP71AkWMlC6n86ilcOXM6vdl0f0z3pTbqkpTBhaC/eVAIqIiJxYsQ9iymY+QILlpXiHCxYVkrBzBcYcc9iv0OTTlKvdnJI7gQ0O6997THi7JNzMUNJQyzxhnOnHSgjxaB7zc6YH87d5FPDevNh5cd8VK2DvYiIxL6ld01h2piBZKYHTmMz01O4fMxAlt49xefIpLPUq50ckjsBnTqbhtTMY9vSu8LU2f7EE6Kcbl0YPTCbtzbt8TsUaRKHw7mbnDssF9AFDRERiQ99e2SSlZFGbX0jGWkp1NY3kpWRRt+szLY3lpikXu3kktwJaOEM/lYwk3LXG4dB9mC4bB4UzvA7sjadOyyX1dv3cuiIqpfGhDgdzg1wev8e9DqpixJQERGJG7sP1nLtxHyeuXUS107Mp/Jgrd8hSSeoVzu5pPkdgN/+d994/jRgAQu/fo7fobTLpFN6M/8fm1m+pYrJI/r6HY5k50H19pbbY1xKivEffVZz8fr5uHv3YNl5gVEAcXAhRkREktP864qOPp87fbSPkUg4xHKvdsX+Gm57fDW/umZsTMSTCJK6B7T6UB3v76jmnFNy/Q6l3cYX9KJLagpvfahhuDFh6mzqUuJvODcAJQu5ruKnDGA3hgsk0nEyf1VEREQSQ6z2amteavi12QNqZo8AnwcqnHPHXWIyMwN+AXwOOATc6Jxb5S1rAN7zVi11zk3z2ocCTwC5wErgOufckc5/nPZ5Z8senINJw3pH+607rWuXVMYOydGwyVhROINH39jMZXv+l/5ud6DnM156EZfMIa2xWQGipvmr8RC/iIicUCKfy0niiLVe7RH3LKY26JaHC5aVsmBZKRlpKWyYe4mPkcW/UHpAHwUuPsHyS4Dh3uMW4DdByw4758Z4j2lB7T8EHnDODQP2Aje3K+oweWvTbjLTUxgzOMePt++0m3qsYP6eG3H35sADo9Vj5aP6hkbmVY7lF2c8A/fugzvfj5/kLY7nr4qISEgeJUHP5WKNbiOSODQvNXLaTECdc28AVSdY5XLgMRfwDpBjZgNaW9m7ynYB8LTX9AdgesgRh9FbH+4JDGVNi8ORyCULuXDT/5BnGjYZC9bu2M+B2vq4HM4dr7cjEhGR0CTyuVys0XDNxBHL81LjXTgyr0FAcPWVMq8NINPMis3sHTOb7rXlAvucc/UtrB81FQdq2FhxMC6H3wKwZA6pDfF5249E1DQX95yT4zABnTo7MF81WLzMXxURkXCIy3O5WKLbiCSmWJ2XGu8iXQU33zlXbmYnA6+a2XtAdXt2YGa3EBgOwpAhQ8IW2NtewnBuPPZYgYZNxpi3N+9heN/u9MnK8DuU9vOGCje8/D1sfzkfZ/Yn61LN/xQREaCT53KROo+LNUvvmsLcRet5ae1H1NQ1kpmewkWj+jPr0tP9Dk06IdbmpSaKcPSAlgODg17neW0455p+bgZeB8YCewgM7Uhrvn5LnHMPOeeKnHNFffr0CUO4AW9t2kOPzDRGDcwO2z6jSsMmY8aR+kZWbKmK34sZAIUzSP2vtUzN+it3Dvyjkk8RkeQSsXO5SJ3HxRoN1xQJXTgS0OeA6y3gbKDaObfTzHqaWQaAmfUGJgHrnHMOeA34grf9DcCzYYijXd7ZsocJQ3NJTbFov3V4aNhkzCgp28fhuob4nP/ZzMShvVi+pYqGRud3KCIiEj1xeS4XazRcUyQ0odyG5XFgMtDbzMqA7wLpAM653wKLCJTt3kSgdPdN3qanA/PNrJFAovsD59w6b9ndwBNmNhdYDTwcrg8Uil37a9i25xDXnZ0fzbcNL6+Hyi35Hm5fOfu79CXnsrnqufLBWx/uwQwmDk2ABPTkXjyxYjsbPjrAyIE9/A5HRETCIBHP5WKRhmuKhKbNBNQ5d3Ubyx3wzRba3wLOaGWbzcCEEGMMu+VbAoXgxhf08iuE8CicgRXO4PqHl1F5oJa/F57nd0RJ6e0P93B6/x70PKmL36F02gQviV62ZY8SUBGRBJGI53IiEr/i8P4jnbd8SxXduqQyKkFOsCcU9OJfHx1g3yHd/znajtQ3sqp0LxNPjvOLGZ5BOV3J69mVZZtPVK1fRERERKRjkjIBXbG1irPye5KWmhgff6J364+mnl2Jnvd3VFNb38iEeO9NDzJxaC7Lt1YRuCAuIiIiIhI+iZGBtcO+Q0f410cHEiphKMzLpktaihJQH6zwfudFCfR9mji0F1UfH2FTxUG/Q/GdmT1iZhVm9n4ry83M5pnZJjMrMbNxXvsYM3vbzNZ67V+KbuQiIiIisSnpEtDirXsBGD80cRKGzPRUxg7OYZkS0KhbsbWKob1Pis/7f7aiaTjxO/o+ATwKXHyC5ZcAw73HLcBvvPZDwPXOuVHe9j83s5zIhSkiIiISH5IuAV2+tYouqSmMGZzjdyhhNXFoL9buqOZATZ3foSSNxkbHiq17GV/Q0+9QwmpIr27075GpHnXAOfcGcKJfxOXAYy7gHQL3xRvgnPvAObfR28cOoAJI3BvgiYiIiIQo+RLQLVUU5mWTmZ7qdyhhNfHkXBodFG/b63coSWNjxUGqD9fFfzXlZsyMCUN7sWzzHs0DbdsgYHvQ6zKv7SgzmwB0AT5saQdmdouZFZtZcWVlZcQCFREREYkFyZOAliyk8Wej+EvF53h4701QstDviMJq7JAc0lJMvVZRtHxr4Hc9IYGGczeZeHIvKg7UsnXPIb9DiWtmNgD4I3CTc66xpXWccw8554qcc0V9+qiTVESkJRX7a5gx/20qDtT4HYqIdFJyJKAlC+H520nZX0aKQfaRj+D52xMqCe3WJY0z8rJZtnmP36EkjRVbquiblcGQXt38DiXsJg5tqqys71MbyoHBQa/zvDbMrAfwAjDLG54rIiIdNG/JRlZsrWLeKxv9DkVEOik5EtAlc6Du8LFtdYcD7QlkwtBelJRVU1PX4HcoSaF4axXjh/bCzPwOJexO6XMSPbulHy3aJa16Drjeq4Z7NlDtnNtpZl2AZwjMD33a3xBFROLXiHsWUzDzBRYsK8U5WLCslIKZLzDinsV+hyYiHZQcCWh1Wfva41RRfi/qGx3vbt/ndyiJrWQh9T8dxT9rruSHpdckVE96EzPjrPyerEzyOcVm9jjwNjDCzMrM7GYz+7qZfd1bZRGwGdgE/A641WufAZwH3Ghma7zHmCiHLyIS95beNYVpYwaSmR44Zc1MT+HyMQNZevcUnyOTztKw6uSV5ncAUZGdB9XbW25PIGflB6qxFm/by8STc32OJkF5w7nT6g6DQfeanYHh3ACFM/yNLczOyu/FK+sr2HOwltzuiXObmfZwzl3dxnIHfLOF9gXAgkjFJSKSLPr2yCQrI43a+kYy0lKorW8kKyONvlmZfocmnRQ8rHruFWf4HY5EUXIkoFNn0/Ds7aQ2BA3DTe8KU2f7F1ME9DqpCyf3OYlVSd5rFVEnGs6dYAlokXd7mZXb9vLZUf19jkZERJLV7oO1XDsxn2smDOHPy0upVI9ZXBtxz2Jq6z+py7dgWSkLlpWSkZbChrmX+BiZREtyJKCFM3ht/S5OW/tzBqXswbLzAslngiUMAEX5PXlp3S4aGx0pKYk3N9F3STKcG+CMQdl0SU1RAioiIr6af13R0edzp4/2MRIJh6V3TWHuovW8tPYjauoayUxP4aJR/Zl16el+hyZRkhwJKPBk7Tls6nEGr317st+hRFRRfi8WFpexefdBhvXN8jucxJMkw7kBMtNTGT2oh+4tKyIiImGjYdWSFEWInHOsLt3L2CE5focScWd5wyZVvTRCps6mIbXZATIBh3M3ub77cuZ9dB3u3hx4YHRCFlwSERGR6GoaVv3MrZO4dmI+lQdr/Q6pXVRAqXOSoge0tOoQuw8eOVqkJ5Gd3Nu7fca2vVw1YYjf4SSewhk8v6ac8R/+ioGW2MO5KVnIZdt+QKp5B9fq7QlbcElERESiJ96HVauAUuckRQLadCuJZEhAdfuMyFvw8UQe6zeBv9w6ye9QImvJHFIbml3ZS9CCSyIiIiJtUQGl8EiKIbgrt+0lKyON4UkyJ/JLme/wx/03a9hkBBypb6SkvJpxQxL/YkYyFVwSERGJFA3XTBy6L214tJmAmtkjZlZhZu+3stzMbJ6ZbTKzEjMb57WPMbO3zWyt1/6loG0eNbMt0bpB+8ptexkzJIfUZKgKW7KQCz64j7yU3Rjuk2GTSkLDYu2Oao7UNyZFb3qrhZUSsOCSiEgiS4RzuXgWPFxT4psKKIVHKENwHwV+BTzWyvJLgOHeYyLwG+/nIeB659xGMxsIrDSzF51z+7zt/ts593QnYg/JgZo6Nuw6wMWjk+Q2EkvmHHu/U9CwyTBaVboPgHHJkIBOnR24eFGX2PfPFRFJAo8Sx+dy8UrDNROT7kvbeW0moM65N8ys4ASrXA485pxzwDtmlmNmA5xzHwTtY4eZVQB9gH2djLld1mzfh3PJMf8T0LDJCFtVupdBOV3p1yMJrnR5FyzqXrqX1AM7ONytPyddogsZIiLxJt7P5eKV7neZmOK9gFIsCMcc0EFA8I0Ry7y2o8xsAtAF+DCo+T5vOMcDZpbR2s7N7BYzKzaz4srKynYHt3LbXsxgzOCcdm8blzRsMqJWb0uO2/kcVTiD1G+tZVzqQu49+UklnyIiiSli53KdPY+LZxquKdKyiBchMrMBwB+Bm5xzTeMQvgOcBowHegF3t7a9c+4h51yRc66oT58+7X7/VaX7GNEvi6zM9PYHH4+mzg4MkwymYZNhsbP6MDuqa5KnN92TkmKcNUSVlUVEklVnzuU6ex4X7+L9fpcikRCO27CUA4ODXud5bZhZD+AFYJZz7p2mFZxzO72ntWb2e+DbYYjjOI2NjtXb9jJtzMBI7D42eT1UDS9/D9tfzseZ/cm6VMMmw2HVtn0AyVEBt5mzCnqy5F8V7DlYS273VgcsiIhIfIrZc7l4p+GaIscLRw/oc8D1XgW1s4Fq59xOM+sCPENgTsExE9S9K2mYmQHTgRarsnXWxoqDHKitT76EoXAGqf+1lgu6P8OdAxco+QyTVaV7yUhL4fQBPfwOJeqK8nsBnxRhEhGRhBKz53Iiknja7AE1s8eByUBvMysDvgukAzjnfgssAj4HbCJQLe0mb9MZwHlArpnd6LXd6JxbA/zJzPoABqwBvh6WT9NM05DBZBsy2eSs/F68vqEC5xyBvw/SGSu37eXMvBy6pCXF7XOPUZiXTVqKsbp0LxeO7Od3OCIi0g7xfC4nIoknlCq4V7ex3AHfbKF9AbCglW0uCDXAzli5bS+5J3UhP7dbNN4u5ozLz+H/VpVRWnWI/NyT/A4nrtXUNbB2RzVf+dRQv0PxRWZ6KiMH9mBVqeaBiojEm3g+lxORxJPQXTmrSvcyLr9n0vb+jR0c6PldrWGTnbZ2RzV1DS75hnMHGTs4h3e3V1Pf0Nj2yiIiIiIiLUjYBHTPwVq27P44aYffAozon0W3LqnqtQqDpuHcyZyAjsvvyeG6BjbsOuB3KCIiIiISpxI2AW0qlpLMCWhqinFmXo56QMNg1bZ9DOnVjT5ZyVsBtin5ViEiEREREemoBE5A95KeapwxKNvvUHw1Lj+H9Tv3c/hIg9+hxC3nXGA495Acv0PxVV7PrvTu3oXV6lEXERERkQ5K2AR05ba9jBqYTWZ6qt+h+Grs4J7UNzreK6/2O5S4Vb7vMBUHahmXxL3pAGbG2CE91aMuIiIiIh2WeAloyULcA6N4YsclPFr9FShZ6HdEvhrr9dppHmjHaf7nJ8YN6cmW3R+z9+MjfociIiIJrGJ/DTPmv03FgRq/QxGRMEusBLRkITx/O1ZdRoo5co58BM/fntRJaG73DPJzu7FqmxLQjlpduo9uXVI5rX+W36H4rumCxurt+j6JiEjkzFuykRVbq5j3yka/QxGRMGvzPqBxZckcqDt8bFvd4UB74Qx/YooB44b05J+bduOcS9pb0nTGqtK9FOZlk5aaWNdrOqIwL5vUFGPVtn1ccFo/v8MREZEEM+KexdTWf3K7rwXLSlmwrJSMtBQ2zL3Ex8hEJFwS64y6uqx97Uli3JAcKg/UUrb3cNsryydKFtL4s1H8tfJS5u++Mal70pt065LGaf2z1AMqIiIRsfSuKUwbM5DM9MApamZ6CpePGcjSu6f4HJl0hoZUS7DESkCz89rXniTGenMXV2/f528g8cQbzp2yv4wUg2wN5z5q3JCerCndR0Oj8zsUERFJMH17ZJKVkUZtfSMZaSnU1jeSlZFG36xMv0OTTtCQagmWWAno1NmQ3vXYtvSugfYkdlr/LDLTUzQPtD1ONJw7yY3Lz+HjIw1srDjgdygiIpKAdh+s5dqJ+Txz6ySunZhP5cFav0OSDhpxz2IKZr7AgmWlOBcYUl0w8wVG3LPY79DER4k1B7RpnueSOYFht9l5geQzied/AqSlplCYl6Me0PbQcO5WjR0c6FFftW0fp/Xv4XM0IiKSaOZfV3T0+dzpo32MRDpr6V1TmLtoPS+t/YiaukYy01O4aFR/Zl16ut+hiY8SKwGFQLKZ5AlnS8YN6cnD/9xMTV1D0t8bNSTZeVC9veX2JJef241eJ3Vhdelerpk4xO9wREREJEZpSLW0JLGG4Eqrxg7Joa7BsXZHtd+hxIeps2lM03DulpgZX++5kv9a929wbw48MFpzY0VERKRFyTCkWkWW2ifxekClRU33b1y1bR9n5ffyN5h4UDiDlVurGFD8Ywal7ME0nPsTJQu5qeoB0p13kK3eHijQBPr9iIiIyDGSYUh1cJGluVec4Xc4MU8JaJLom5VJXs+uun1GOzzbOIm/2lDenf1ZUlN0/9SjlswhvbHZFT7db1dERESSjO5b2zEagptEvpZdzP/b+CUNmwzR6tJ9nDk4W8lncyrQJCIiIqL71naQEtBkUbKQayt+ygAqAffJsEkloS06dKSef310gHHePVQliO63KyIi0imaM5gYVGSpY0JKQM3sETOrMLP3W1luZjbPzDaZWYmZjQtadoOZbfQeNwS1n2Vm73nbzDMzdTNF0pI5pLU2bFKO8+72ahoanRLQliTR/XYjceyLipKFgVEOkRrtEMn9K3Z/9q/Y/dt/iMws28y+ZGbf8h5fMrOcELfVeVyMCZ4zKPGts0WWml+MaOniRCht0d6uM0LtAX0UuPgEyy8BhnuPW4DfAJhZL+C7wERgAvBdM2s6o/8N8LWg7U60f+ksDZtsl6a5smMG5/gbSCwqnAGXzePjrgNodEZd90Fw2bxEnf/5KOE/9kVWycLA6Ibq7URktEMk96/Y/dm/Yvdv/yEys+uBVcBkoJv3mAKs9Ja15VF0HhcTRtyzmIKZL7BgWSnOBeYMFsx8gRH3LPY7NOmg+dcVMXf6aEYO7MHc6aOPKboUiuYXI1q6OBFKW7S36wxzzoW2olkB8Dfn3HHlq8xsPvC6c+5x7/UGAgfJycBk59y/B6/nPV5zzp3mtV8dvF5rioqKXHFxcUjxSjMPjG7lvpaD4c4WL4gmta/+oZjNlQd59duT/Q4lZm2uPMgFP/0HP/y3M/jS+PDfD9TMVjrn2ncUj4BwHvua1mtNqMe47z2/lnU79re47Fe7rqdPY8Vx7ZUpfbmt32Nt7rstkdy/Yvdn/4o9PPsfObAH371sVMj7bs8xzju2THTO7WvW3hNY5pw7NYR9FKDzON9V7K9h7qL1vLT2I2rqGslMT+GiUf2ZdenpGraZZJoXMIo3bRVaOtExLlxzQAcBwdlNmdd2ovayFtqPY2a3mFmxmRVXVlaGKdwklETDJjvLOcea7XsZ4926Rlo2tPdJ5HRLZ9W2fX6H4qf2HvuOE+5jXG5jy/torT2W9q/Y/dm/Yvdv/+1gQEs9Bo3ess7SeVyUaM6gNGlewCgjzRiU05WMtE8KGl00qh8Xjep3TJGj5m3R3i4chZZi/jYszrmHgIcgcOXM53Dilzc88vDi75JxaCf13QfS5aJ7E3XYZKdsrzrM7oNHNP+zDWbG2ME5urVPJ3XkGHfCXpYH8loc7ZCSnceT/35Ox4KM1v4Vuz/7V+z+7T909wGrzOwlPkkIhwAXAt+PZiDtpfO44zXNGbxmwhD+vLyUShUiSkrNL0YcaWikW5dUjjR8cnGiT/cMHBxzwaJ5W7S3C8dFk3D1gJYDg4Ne53ltJ2rPa6FdIqlwBvu/sZqTa//EH895QclnK1aVBhIqJaBtGzekJxsrDrK/ps7vUPzS3mNf5EV6tEMk96/Y/dm/Yvdv/yFyzv0BGA/sBOqAGgLDYIucc4+G4S10HhdFnZ0zKImjeQGj6sN1xxU0aqnIkd/btbfQUnPhmgN6KXAb8DkCE9XnOecmeJPXVwJN1dRWAWc556rMbDlwO7AMWAT80jm36EQxaO5AeJx7/xLG5ffkV9eMa3vlJPTdZ9/nqZVllHz3s6Sl6k5FJ/LPjbv58sPL+OPNE/j08D5h3XeczAFt97HvRO8VtmNcycJAhevqssDtcabODu8Fp0juX7H7s3/FHvX9d+QYZ2ZzgasIHFMeAV50IZ7I6TxORKLpRMe4kBJQM3ucwET03sAuAhXR0gGcc7/1Sm//ikAFtEPATc65Ym/brwD/n7er+5xzv/faiwhUZesKLAb+o62DqA5c4fHNP69iTek+3px5gd+hxKTLfvlPumek8fgtZ/sdSsw7UFNH4fde4j+nnsodnxke1n3HQgIaiWPfiegYJ5I8OnqM8447nwVuAoqAhcDDzrkPT7CNzuNEJKpOdIwLaQ6oc+7qNpY74JutLHuEwFW65u3FwHFX4STyxg7O4YWSnezaX0O/Hpr0HuzwkQbW79zPv59/st+hxIWszHRG9Ms6Omw50UTi2Cci0hnOOWdmHwEfAfVAT+BpM3vZOXdXK9voPE5EYobGFyahcfmBuY2rEzRp6Iz3yqupb3SMHaz5n6EaO6Qnq0v30tio2hIiIpFkZneY2UrgR8CbwBnOuW8AZwH/5mtwIiIhUgKahEYN7EGX1BRWl+7zO5SY09STN1a3YAnZuCE57K+pZ/Pug36HIiKS6HoBVzrnLnLOPeWcqwNwzjUCn/c3NBGR0CgBTUIZaamMGtSDldvUA9rc6tK95Od2I7d7ht+hxI2mHnV9n0REIss5913n3LZWlq2PdjwiIh2hBDRJnTWkJyXl1Rypb/Q7lJjhnGNV6T7dfqWdTu59Ejnd0lm1bZ/foYiIiIhIjFMCmqTG5ffkSH0j63bu9zuUmFG29zCVB2oZp+G37WJmjBvSM2ELEYmIiIhI+CgBTVJnadjkcVZv3wcEiupI+4wbksPGioNUH6rzOxQRERERiWFKQJNUvx6ZDMrpql6rIKu27SUzPYXT+mf5HUrcOVpZebu+TyIiIiLSOiWgSWxcfk9WqQf0qNWleynMyyEtVf8t2uvMvBxSDFapsrKIiIiInIDOtJPYuCE57KyuYce+w36H4q+ShTT+bBTPVF7K7/bcCCUL/Y4o7pyUkcZp/XvogoaIiIhQsb+GGfPfpuJAjd+hSAxSAprEmuaBJvUw3JKF8PztpOwvI8Ug+8hH8PztSkI74Kz8nqzZvo+GRud3KCIiIuKjeUs2smJrFfNe2eh3KBKDlIAmsdMH9CAzPSW5b5+xZA7UNesBrjscaJd2GZefw8HaejZWHPA7FBEREfHBiHsWUzDzBRYsK8U5WLCslIKZLzDinsV+hyYxRAloEktPTaEwL4eVydwDWl3WvnZpVdP9U1VZWUREJDktvWsK08YMJDM9kGJkpqdw+ZiBLL17is+RSSxRAprkxg3pybod1dTUNfgdij+y89rXLq0aUv4Cb2XezjWLz4QHRmsYs4iISJLp2yOTrIw0ausbyUhLoba+kayMNPpmZfodmsQQJaBJ7qz8ntQ1ON4rr/Y7FH9MnU1jWtdj29K7wtTZ/sQTr0oWYs/fzkB2Yzio3q65tCIiIklo98Farp2YzzO3TuLaiflUHqz1OyRfqBBT69L8DkD8NXZIDhC4B+b4gl7+BuOHwhmsKd1L3+U/YlDKHiw7L5B8Fs7wO7L4cqK5tPpdioiIJI351xUdfT53+mgfI/FXcCGmuVec4Xc4MUUJaJLr3T2D/NxuST1v72/u0/ypcQjvzb6ILmkaFNAhmksrIiIiwoh7FlNb33j09YJlpSxYVkpGWgob5l7iY2SxQ2fbwteyi/ne5i/h7s1Jyrl7q7fvpTAvW8lnZ2gurYiISIdpuGbiUCGmtumMO9mVLOSqj37CgCSdu1db38Da8v1HK7hKB02dHZg7G0xzaUVEREKi+2YmDhVialtICaiZXWxmG8xsk5nNbGF5vpktMbMSM3vdzPK89ilmtiboUWNm071lj5rZlqBlY8L5wSRES+aQ1tjsalsS3Qfz/fJqjjQ0MlYJaOcUzoDL5kH2YBxGuetNw6W/0PxPEZEYoPO42KX7ZiYmFWI6sTbngJpZKvAgcCFQBqwws+ecc+uCVvsJ8Jhz7g9mdgFwP3Cdc+41YIy3n17AJuCloO3+2zn3dFg+iXRMks/dK94amPt6Vr4S0E4rnAGFM3j+3R3c/vhq/tbnUyRv6QERkdig87jYtvSuKcxdtJ6X1n5ETV0jmekpXDSqP7MuPd3v0KQTVIjpxELpAZ0AbHLObXbOHQGeAC5vts5I4FXv+WstLAf4ArDYOXeoo8FKBCT53L3ibXspyO1Gn6wMv0NJGOOaKiuXJm9hKxGRGKLzuBim4ZqSjEJJQAcB24Nel3ltwd4FrvSeXwFkmVlus3WuAh5v1nafN9zjATNrMQMws1vMrNjMiisrK0MIV9oliefuOedYtW0vZ+Un4e1nImhQTlf6ZmWwKokrK4uIxBCdx8U4DdeUZBOuIkTfBs43s9XA+UA50NC00MwGAGcALwZt8x3gNGA80Au4u6UdO+cecs4VOeeK+vTpE6Zw5Shv7t7hbgNpdEbNSQMDc/mSYO7elt0fs+fjIxQVaPhtOJkZZ+X3ZFXpPr9DERGR0Og8zkfzryti7vTRjBzYg7nTRx8zfFMkEYWSgJYDg4Ne53ltRznndjjnrnTOjQVmeW37glaZATzjnKsL2manC6gFfk9giIj4oXAGR24r4eTaP/Fw0fNJkXxCYPgtQJHmf4bduCE9Ka06ROUBXcUVEfGZzuNEJKaEkoCuAIab2VAz60JgCMZzwSuYWW8za9rXd4BHmu3japoN2/CupmFmBkwH3m939BI22d3SGda3OyuTaNjkyq17ye6azil9uvsdSsIZl58DaB6oiEgM0HmciMSUNhNQ51w9cBuBYRfrgYXOubVmNsfMpnmrTQY2mNkHQD/gvqbtzayAwJW3fzTb9Z/M7D3gPaA3MLdzH0U6a9yQHFaV7sU553coUVG8rYqz8nuSkmJ+h5JwRg3MpktqihJQERGf6TxORGJNm7dhAXDOLQIWNWubHfT8aaDFMtzOua0cP9kd59wF7QlUIu+s/J4sLC5j8+6PE75XsOrjI3xY+TFXjkuOar/RlpmeyqhBPVSISEQkBug8TkRiSbiKEEkCGDckMBcyGYbhrtT8z4g7a0hP3i2rpra+oe2VRURERCQpKAGVo07p052cbukUb63yO5SIK95WRXqqcebgHL9DSVjjh/biSH0j75VV+x2KiIiIiMQIJaByVEqKUZTfkxVbk6AHdOteRg3MJjM91e9QElZT73IyfJ9EREREJDRKQOUY4wt6sWX3x1QcqPE7lIiprW+gpLxaw28jLLd7Bqf0OYkVSdCjLiIiIiKhUQIqxxg/tBcAxQnca/V+eTVH6hspKlACGmnjC3pRvLWKxsbkqKwsIiIiIiemBFSOMXpgNl3TU1m+JXF7rZqS67Pye/kcSeIbX9CL/TX1fFBxwO9QRERERCQGKAGVY3RJS2HskJyEHjZZvG0v+bnd6JOV4XcoCW98QSDJX5HAFzREREREJHRKQOU44wt6sW7nfvbX1PkdSniVLMQ9MJr5mz/DM7X/DiUL/Y4o4Q3u1ZV+PTLithCRmV1sZhvMbJOZzWxheb6ZLTGzEjN73czygpb9yMzWmtl6M5tnZhbd6EVERCKvYn8NM+a/ndD1QyS8lIDKcSYM7YVzCXY/0JKF8PztWPV2UnD0qt8Fz9+uJDTCzIzxBb1YsbUK5+JrHqiZpQIPApcAI4GrzWxks9V+AjzmnCsE5gD3e9ueC0wCCoHRwHjg/CiFLiIiEjXzlmxkxdYq5r2y0e9QJE4oAZXjjB2SQ1qKJdawySVzoO7wsW11hwPtElHjC3qxs7qGsr2H2145tkwANjnnNjvnjgBPAJc3W2ck8Kr3/LWg5Q7IBLoAGUA6sCviEYuIiETJiHsWUzDzBRYsK8U5WLCslIKZLzDinsV+hyYxTgmoHKdblzRGDcpOrHmg1WXta5ewaZoHWrwt7r5Pg4DtQa/LvLZg7wJXes+vALLMLNc59zaBhHSn93jRObe+pTcxs1vMrNjMiisrK8P6AURERCJl6V1TmDZmIJnpgXQiMz2Fy8cMZOndU3yOTGKdElBp0YSCnry7vZqauga/QwmP7Lz2tUvYnFb5d97KuJ3pz46GB0Yn2rDnbwPnm9lqAkNsy4EGMxsGnA7kEUhaLzCzT7e0A+fcQ865IudcUZ8+faIVt4iISKf07ZFJVkYatfWNZKSlUFvfSFZGGn2zMv0OLWZpvmyAElBp0fiCXhxpaKSkrNrvUMJj6mwa07oe25beFabO9ieeZFGykJS/3c5A243hoHp7PM29LQcGB73O89qOcs7tcM5d6ZwbC8zy2vYR6A19xzl30Dl3EFgMnBOVqEVERKJk98Farp2YzzO3TuLaiflUHqz1O6SYpvmyAWl+ByCx6ejtM7ZWMWFoAtwvs3AGyzbvYfCqnzAoZQ+WnRdIPgtn+B1ZYjvR3NvY/92vAIab2VACiedVwDXBK5hZb6DKOdcIfAd4xFtUCnzNzO4HjEDv6M+jFLeIiEhUzL+u6OjzudNH+xhJbBtxz2Jq6xuPvl6wrJQFy0rJSEthw9xLfIzMH+oBlRb1PKkLt+QU86V/XgL35iTE0MmnjpzD5em/he/uhTvfj4cEKP7F8dxb51w9cBvwIrAeWOicW2tmc8xsmrfaZGCDmX0A9APu89qfBj4E3iMwT/Rd59zz0YxfRETim4ZrJg7Nlz2WekClZSUL+Xbtg3Rx3lCKpqGTELeJ27LNgd5c3Y4xirLzAt+dltrjgHNuEbCoWdvsoOdPE0g2m2/XAPx7xAMUEZGEFTxcc+4VZ/gdjnSC5sseSwmotGzJnE+SzybxM3TyOGV7D1G+7zBf+/RQv0NJLlNnBy5cBA/D1dxbERGRVmm4ZmJqmi97zYQh/Hl5KZVJ3LMdUgJqZhcDvwBSgf91zv2g2fJ8AnOf+gBVwJedc2XesgYCw9AASp1z07z2oQTuq5cLrASu8+61J7EgjodOtmS5d0/TCUNzfY4kyTRdrFgyJ/Dd0dxbERGRE1p61xTmLlrPS2s/oqaukcz0FC4a1Z9Zl57ud2jSCZov+4k254CaWSrwIHAJgZuuX21mI5ut9hPgMedcITAHuD9o2WHn3BjvMS2o/YfAA865YcBe4OZOfA4JtwS7bcmyzVVkd03ntP5ZfoeSfApnBObc3rtPc29FRHxgZheb2QYz22RmM1tYnm9mS8ysxMxeN7O8oGUNZrbGezwX1D7UzJZ5+3zSzLpE6/MkOg3XlEQXShGiCcAm59xmr4fyCeDyZuuMBF71nr/WwvJjWGAS3gV8MnfqD8D0EGOWaJg6OzBUMlgcD51ctmUP4wt6kZKi+Z8iIpI81JEQn3R7E0lkoQzBHQQEVxEpAyY2W+dd4EoCw3SvALLMLNc5twfINLNioB74gXPurwSG3e7zqkw27XNQhz+FhJ/XS3Vo8XfJPLSTuu4Dybjo3rjsvdq1v4atew7x5bPz/Q5FREQk2o52JACYWVNHwrqgdUYC3/Kevwb89UQ7DOpIaLo11R+Ae4HfhCvoZKfhmpLIwnUblm8D55vZagL3uysHGrxl+c65IgIHqZ+b2Snt2bGZ3WJmxWZWXFlZGaZwJSSFM6i57V1Orv0Tvzvr2bhMPgHe+nA3AGefrPmfIiKSdFrqSGh+0b+pIwGCOhK815needg7Zjbdawu5I0HncSLSXCgJaDkwOOh1ntd2lHNuh3PuSufcWGCW17bP+1nu/dwMvA6MBfYAOWaW1to+g/b9kHOuyDlX1KdPnxA/loRLr5O6cPqAHrz14R6/Q+mwNzftIadbOiMH9PA7FBERkVgUsY4EnceJSHOhJKArgOHeZPMuwFXAc8ErmFlvM2va13cIVMTFzHqaWUbTOsAkYJ1zzhEY4vEFb5sbgGc7+2EkMs45OZeV2/ZSU9fQ9soxxjnHm5t2c+4puZr/KSIiycjXjgQRkebaTEC94RW3AS8C64GFzrm1ZjbHzJomo08GNpjZB0A/4D6v/XSg2MzeJZBw/sA51zTn4G7gW2a2icBQjofD9JkkzM49JZfa+kZWl+7zO5R227L7Y3ZW1zBpWG+/QxEREfGDOhJEJKaEdB9Q59wiYFGzttlBz5/mk4q2weu8BZzRyj43E5gYLzFuwsm9SDF4e/MezjklvuZRvrkpMP9z0ilKQEVEJPk45+rNrKkjIRV4pKkjASh2zj1HoCPhfjNzwBvAN73NTwfmm1kjgU6L5h0JT5jZXGA16kgQkRCFlIBKcuuRmc4Zg7J5+8PdcOGpfofTLm9u2sOgnK7k53bzOxQRERFfqCNBRGJJuKrgSoI755TerC7dx8e19W2vHCMaGh1vfbibScNyCVSMFxERERERPykBlZB8alhv6hsdy7bETzXctTuq2V9Tr/mfIiIiIiIxQgmohKSooCeZ6Sm88cFuv0MJ2ZubAsnyuZr/KSIiIiISE5SASkgy01M5++Rc/vFBHNxEumQhPDCar79+Fsu63kGfLSrMJyIiItJZFftrmDH/bSoO1PgdisQxJaASsvOG92HL7o/ZXnXI71BaV7IQnr8dqrdjOPq5ysDrkoV+RyYiIiIS1+Yt2ciKrVXMe2Wj36FIHFMVXAnZ+SP6wN/gHx9U8uWz8/0Op2VL5kDd4WPb6g4H2gtn+BOTiIiISBwbcc9iausbj75esKyUBctKyUhLYcPcS3yMTOKRekAlZCf3PolBOV1jexhudVn72kVERETkhJbeNYVpYwaSmR5IHTLTU7h8zECW3j3F58gkHikBlZCZGeed2oe3P9xDXUNj2xv4ITuvfe0iIiIickJ9e2SSlZFGbX0jGWkp1NY3kpWRRt+sTL9DkzikBFTa5fxT+3Cwtp5V2/b6HUrLps6mMa3rsW3pXWHq7JbXFxEREZE27T5Yy7UT83nm1klcOzGfyoO1fockcUpzQKVdzqt9jTcz7mHgY3sCvYpTZ8fW3MrCGfxjQyXD3/sZg1L2YLEYo4iIiEgHVOyv4bbHV/Ora8ZGvfdx/nVFR5/PnT46qu8tiUUJqISuZCHd/n4n3cwr8lO9PVBhFmIqwXtkfxE7sh9myX9N9jsUERERkbAJrkI794oz/A5HpEOUgEro4qDC7KEj9SzbXMV158RolV4RERGRdlIVWkkkmgMqoYuDCrNvf7iHIw2NTBnR1+9QRERERMJCVWglkSgBldDFQYXZV9bv4qQuqYwf2tPvUERERETCQlVoJZEoAZXQTZ0dqCgbxKXFToXZhkbHy+t2Mfm0vmSkpfodjoiIiEjYqAqtJArNAZXQNc3zXDIHV11GeWMuFWPuYlyMzP9cXbqX3QePcNGo/n6HIiIiIhJWqkIriUIJqLRP4QwonEFjo2Pafa/w6YO9Ged3TJ4X135EeqoxeUQfv0MREREREZEWhDQE18wuNrMNZrbJzGa2sDzfzJaYWYmZvW5meV77GDN728zWesu+FLTNo2a2xczWeI8xYftUEnGpKcYFp/XltX9VUNfQ2PYGEeac46V1uzj3lN70yEz3OxwREREREWlBmwmomaUCDwKXACOBq81sZLPVfgI85pwrBOYA93vth4DrnXOjgIuBn5tZTtB2/+2cG+M91nTqk0jUXTiyH/tr6lm+pcrvUNiw6wDb9hzS8FsREZFm1JEgIrEklCG4E4BNzrnNAGb2BHA5sC5onZHAt7znrwF/BXDOfdC0gnNuh5lVAH2AfZ0NXPx33vA+dOuSygvv7WTSsN5Rec+6ujrKysqoqak5pn1/TR3/O20A/bsfZP369VGJRcIjMzOTvLw80tPVcy0iEm5BHQkXAmXACjN7zjkXfB7X1JHwBzO7gEBHwnV80pGw0cwGAivN7EXn3D5vu/92zj0dtQ8jIgkhlAR0ELA96HUZMLHZOu8CVwK/AK4Assws1zm3p2kFM5sAdAE+DNruPjObDSwBZjrnjivnZWa3ALcADBkyJIRwJVq6dknlM6f3Y/F7O/netFGkp0a+qHJZWRlZWVkUFBRgZkfbP9h1gFwzhvXtHvEYJHycc+zZs4eysjKGDh3qdzgiIolIHQkiElPClTF8GzjfzFYD5wPlQEPTQjMbAPwRuMk51zRh8DvAacB4oBdwd0s7ds495Jwrcs4V9emj4jKx5rIzB7L3UB1vbtodlferqakhNzf3k+TzUBWNH73P8PpNDG3YCof8Hw4soTMzcnNzj+vRFhGRsGmpI2FQs3WaOhIgqCMheIUTdCSUmNkDZpbR0pub2S1mVmxmxZWVlZ35HCKSIEJJQMuBwUGv87y2o5xzO5xzVzrnxgKzvLZ9AGbWA3gBmOWceydom50uoBb4PYErdBJnzju1N1mZaTz/7s6ovWdw8kn1dlIa6zCDVFcH1duVhMaZ4J7sWNPReVPesiFm9pKZrTezdWZWENXgRURCp44EaVHF/hpmzH+bigO6UCzhE0oCugIYbmZDzawLcBXwXPAKZtbbzJr29R3gEa+9C/AMgXkFTzfbZoD304DpwPud+Bzik4y0VC4e1Z+X1n5ETV1D2xuE04Gd4JpV4HWNgXaRTupkATaAx4AfO+dOJ3CBrSLyUYuIHEcdCdJh85ZsZMXWKua9stHvUCSBtJmAOufqgduAF4H1wELn3Fozm2Nm07zVJgMbzOwDoB9wn9c+AzgPuLGFKml/MrP3gPeA3sDcMH0mibLLzhzIgdp6/vFBlIfWNBxpX3sYfeUrX6Fv376MHt2+G0Hff//9DBs2jBEjRvDiiy+2uM6vfvUrhg0bhpmxe/cnQ5udc2zdupVHH330hO+xY8cOvvCFL7QZi3OOCy64gP379wOtf6aqqiouvPBChg8fzoUXXsjevXvbFU/TZ3rkkUfaXC/GHJ035Zw7AjTNmwo2EnjVe/5a03IvUU1zzr0M4Jw76Jw7FJ2wRUSOoY4EabcR9yymYOYLLFhWinOwYFkpBTNfYMQ9i/0OTRJASHNAnXOLnHOnOudOcc7d57XNds495z1/2jk33Fvnq03FhJxzC5xz6UG3Wjl6uxXn3AXOuTOcc6Odc192zh2M0GeUCDv3lFx6ndSFZ9eUt71yOKV2aV97GN144438/e9/b3V5QUHBcW3r1q3jiSeeYO3atfz973/n1ltvpaHh+F7jSZMm8corr5Cfn39M+9e//nX++c9/Ulpays0330x5ecu/74EDB/L0020XJVy0aBFnnnkmPXr0OOFn+sEPfsDUqVPZuHEjU6dO5Qc/+EG74oFAcvvLX/6yzZhiTGfmTZ0K7DOzv5jZajP7sdejKiISVepIkI5YetcUpo0ZSGZ6IFXITE/h8jEDWXr3FJ8jk0QQShVckRNKS01h1uD3OPuDB3H37sGy82DqbCicEdH3/d5bNawra2G+Z1ompLzdoX2OHNiD7142qs31zjvvPLZu3dqufT/77LNcddVVZGRkMHToUIYNG8by5cs555xzjllv7NixLW7/61//mmnTprF27VqWL19O3759+cc//sEdd9wBBOZSvvHGG+zZs4fPf/7zvP/++zz66KM899xzHDp0iA8//JArrriCH/3oRwD86U9/4pZbbmnzMz377LO8/vrrANxwww1MnjyZH/7why3Gc8cdd5Cbm8vs2bN58cUXue+++3j99dfp1q0bBQUFLF++nAkTEmqU1reBX5nZjcAbfDJvKg34NDAWKAWeBG4EHm6+A1X6FpFIc84tAhY1a5sd9Pxp4Lgrl865BcCCVvZ5QZjDlBjSt0cmWRlp1NY3kpGWQm19I1kZafTNyvQ7NEkAkb9vhiS+koVcUfYjBtluDBcoBPT87VCyMKJve4Qu1Lp0wCtiY+Yln7F5XaW8vJzBgz+ZhpOXl3fCXsPmbrvtNq6++mq+8pWvMGvWLHbs2MFPfvITHnzwQdasWcPSpUvp2rXrcdutWbOGJ598kvfee48nn3yS7dsDnXpvvvkmZ511Vpvvu2vXLgYMGABA//792bVrV6vx3H///Tz55JO89tpr3H777fz+978nJSVwmCkqKmLp0qUhf94Y0Jl5U2XAGm/4bj2BWxqMa+lNVKBDREQ6KpJFgnYfrOXaifk8c+skrp2YT+XB4+6WKNIhsXmmLvFlyRxS6g8f21Z3GJbMiWgv6Fc/fTI1dfmc1j8rJiqp3nfffTz11FNAYB7mmDFjgMCQ2gcffLDT+//1r3/Ntm3bqK+vZ/bs2Uf3/a1vfYtrr72WK6+8kry8vOO2mzp1KtnZ2QCMHDmSbdu2MXjwYKqqqsjKympXDGZ29HfdUjwAv/vd7zjvvPN44IEHOOWUU4629+3bl3/961/t/tw+OjpvikDieRVwTfAKZtYbqPKqQh6dN+Vtm2NmfZxzlcAFQHHUIhcRkaQQXCRo7hVnhHXf868rOvp87vT21bwQOREloNJ51WXtaw+DI/UNHKipo1+PzJhIPgFmzZrFrFmzgMAc0DVr1hyzfNCgQUd7HwHKysoYNKj5lMLWmRkFBQXceOONR9tmzpzJpZdeyqJFi5g0aRIvvvgimZnHDo/JyPjk1mypqanU19cDkJaWRmNj49Eeytb069ePnTt3MmDAAHbu3Enfvn1bjQfgvffeIzc3lx07dhzTXlNT02IPbaxyztWbWdO8qVTgkaZ5U0CxNwd+MnC/mTkCQ3C/6W3bYGbfBpZ4BTpWAr/z43OIiEjiGXHPYmrrP7kTwIJlpSxYVkpGWgob5l7iY2QibdMQXOm87ON73U7YHgZVH9cB0LNb5AsOhcu0adN44oknqK2tZcuWLWzcuLHT8yE//PBDzjjjDO6++27Gjx/frh7GESNGsHnz5jbXmzZtGn/4wx8A+MMf/sDllzcvBPuJbdu28dOf/pTVq1ezePFili1bdnTZBx980O6qwX7raAE2b9nLzrlCr9jajV4lXRERkU5TkSCJZ0pApfOmzob0Y3u2XFrXQHsENDY6qj4+Qo/MdLqk+fMVvvrqqznnnHPYsGEDeXl5PPzwcbVljjNq1ChmzJjByJEjufjii3nwwQdJTQ0URv3c5z53tMdw3rx55OXlUVZWRmFhIV/96ldb3efPf/5zRo8eTWFhIenp6VxySehXPS+99NKjxYVO9JlmzpzJyy+/zPDhw3nllVeYOXNmi/tzznHzzTfzk5/8hIEDB/Lwww/z1a9+lZqawLyUN998kwsvvDDk+ERERKRlKhIk8cycc37HELKioiJXXKxpVDGpZCEsmYOrLqO8MZf3T7+Di6++Pexvs379evoNOYWyvYc4ufdJdM9MD/t7JIudO3dy/fXX8/LLL0f8vVavXs3PfvYz/vjHP7a4fP369Zx++unHtJnZSudcUYsbJCgd40SSh45x0ln//sdi+mRlcs2EIfx5eSmVB2qOmbcp4qcTHeM0B1TCo3AGFM7AgFmPLGfd5v1MqW8gIy38tz7cfbCWzPRUTsrQ17czBgwYwNe+9jX2799/9F6gkbJ7926+//3vR/Q9REREkomKBEm80hBcCbuvfnoolQdqeW7NjrZXbqeaugZq6hro3T0jZooPxbMZM2ZEPPkEuPDCCykoKIj4+4iIiIhIbFMCKmH3qWG9OX1AD379+ofUNzS2vUEoShbiHhhNxsflnJaynRw7GJ79ioiIiIhI1CgBlbAzM+78zHC27P6YZ1aXd36HJQvh+dux6u0Y0IV6Uqq3w6Gqzu9bRERERKjYX8OM+W9TcaDG71AkwSkBlYi4cGQ/zhiUzbxXN3KkvpO9oEvmQN3hY9tcIxzY2bn9ioiIiAgA85ZsZMXWKua9stHvUCTBKQGViDAz/uuzpzJ238vU/ngk3JsDD4wO9Ga2V3VZy+0Nuq2iiIiISLD29mSOuGcxBTNfYMGyUpyDBctKKZj5AiPuWRzhSCVZKQGViJlc+zo/zniYrNqdgIPq7fD87e1OQht7DGp5QWqXtjcuWRhIfDuTAIuIiIjEifb2ZC69awrTxgwkMz2QFmSmp3D5mIEsvXtKJMOUJKYEVCJnyRwyXO2xbXWHA0Nq2+GZnjdzyDVLNi0FsgaceENv7ijV2+lMAhwNzjluv/12hg0bRmFhIatWrfI7JBEREYkjHe3J7Nsjk6yMNGrrG8lIS6G2vpGsjDT6ZmVGKXJJNkpAJXJaGzrbWnsL3vigkv/aMIJFBd+B7MGBxtQugefdep1445bmjnYgAY6GxYsXs3HjRjZu3MhDDz3EN77xDb9DEhERkTjSmZ7M3QdruXZiPs/cOolrJ+ZTebC2zW1EOirN7wAkgWXneb2PLbSHYGf1Yf7rqXcZ3rc7n//yHZD+LVi/HvqdHtr7hyEBbs1jjz3GT37yE8yMwsJC/vjHPx63TmVlJV//+tcpLS0F4Oc//zmTJk1qcX/PPvss119/PWbG2Wefzb59+9i5cycDBrTRyysiIiJC53oy519XdPT53OmjIxmmiBJQiaCpswNDXoN6IQ+TwZFzv0N2a9uULIQlc3DVZaRYb6Y0XsXNN99NZnpq+9+/kwlwa9auXcvcuXN566236N27N1VVLd8O5o477uDOO+/kU5/6FKWlpVx00UWsX7++xXXLy8sZPHjw0dd5eXmUl5crARUREZGQNfVkXjNhCH9eXkplK4WIKvbXcNvjq/nVNWM11FaiLqQhuGZ2sZltMLNNZjazheX5ZrbEzErM7HUzywtadoOZbfQeNwS1n2Vm73n7nGdmFp6PJDGjcAZcNs8bOmsc6T6I2Y1f47J/DGTXm48dXxwoaM6m4ejnKrk/7XeMqOhgFbapsyG967Ft6V0D7Z3w6quv8sUvfpHevXsD0KtXy0OBX3nlFW677TbGjBnDtGnT2L9/PwcPHuzUe4uIiLSXzuOSx/zripg7fTQjB/Zg7vTRR3s2m1fG1S1XxE9t9oCaWSrwIHAhUAasMLPnnHPrglb7CfCYc+4PZnYBcD9wnZn1Ar4LFAEOWOltuxf4DfA1YBmwCLgYUL3nRFM4I/AAugDXlO4l/fc/o8dLvwXzbqPiFQdyaV2xZnM2UxtqAnM2vX20+70hsH11WaDnc+rsju2rAxobG3nnnXfIzGz7yuKgQYPYvv2T3tqysjIGDWql+q+IiEiIdB4n8EnCec7/LKHBfdK+YFkpC5aVkpGWwoa5l/gXoCSVUHpAJwCbnHObnXNHgCeAy5utMxJ41Xv+WtDyi4CXnXNV3sHqZeBiMxsA9HDOveOcc8BjwPTOfRSJB2OH9GRO97/Q1Zrdw7PuMBxueShrp+ZsFs6AO9+He/cFfoYh+bzgggt46qmn2LNnD0CrQ3A/+9nP8stf/vLo6zVr1rS6z2nTpvHYY4/hnOOdd94hOztbw29FRCQcdB6XxJpXxg1OPkG3XBF/hJKADgKCJ9KVeW3B3gWu9J5fAWSZWe4Jth3kPT/RPgEws1vMrNjMiisrK0MIV2Jd2oHy9m3QyTmb4TZq1ChmzZrF+eefz5lnnsm3vvWtFtebN28excXFFBYWMnLkSH7729+2us/Pfe5znHzyyQwbNoyvfe1r/PrXv45U+CIiklx8PY8Tf7VUGbcgtxtm6JYr4ptwFSH6NvArM7sReAMoBxrCsWPn3EPAQwBFRUWujdUlHrRSHMi69oL6w8feOiUMczYj4YYbbuCGG2444Tq9e/fmySefDGl/ZsaDDz4YjtBERETaK2LncWZ2C3ALwJAhQ8KxS2mHlirjNjS6kAoViURKKAloOTA46HWe13aUc24H3pUzM+sO/Jtzbp+ZlQOTm237urd9XrP2dnaLSdxqoTou6V3hkh8Gnvs0Z1NERCQB+Xoep44E/7VUGbfpViu65Yr4IZQEdAUw3MyGEji4XAVcE7yCmfUGqpxzjcB3gEe8RS8C/2NmPb3XnwW+45yrMrP9ZnY2gcnr1wO/RJJDW8WB4izhvO+++3jqqaeOafviF7/IrFmzjlv397//Pb/4xS+OaZs0aZJ6P0VEJFJ0HpfkdI9PiTVtJqDOuXozu43AQSgVeMQ5t9bM5gDFzrnnCFwdu9/MHIGhG9/0tq0ys+8TOPgBzHHONVVsuRV4FOhKoGqaKqclk6DquO3lnCOWqr3PmjWrxWSzJTfddBM33XRThCOKL4H6FSIiEgk6jxORWGPxdPJXVFTkiouL/Q5DfLRlyxaysrLIzc2NqSRUOsY5x549ezhw4ABDhw49ZpmZrXTOFbWyaULSMU4keegYJyKJ7ETHuHAVIRKJiry8PMrKylBF5MSRmZlJXl5sVToWERERkchQAipxJT09/bieMhERERERiQ+h3AdUREREREREpNOUgIqIiIiIiEhUKAEVERERERGRqIirKrhmVglsO8EqvYHdUQon3BS7PxS7P0KJPd851ycawcQKHeNilmL3R6LHrmNc2+L1OxCvcYNi90u8xn6iuFs9xsVVAtoWMyuO15Lmit0fit0f8Ry7n+L596bY/aHY/RHPsceSeP09xmvcoNj9Eq+xdzRuDcEVERERERGRqFACKiIiIiIiIlGRaAnoQ34H0AmK3R+K3R/xHLuf4vn3ptj9odj9Ec+xx5J4/T3Ga9yg2P0Sr7F3KO6EmgMqIiIiIiIisSvRekBFREREREQkRikBFRERERERkaiIywTUzC42sw1mtsnMZrawPMPMnvSWLzOzAh/CbFEIsX/LzNaZWYmZLTGzfD/ibElbsQet929m5swsZspJhxK7mc3wfvdrzezP0Y6xNSF8Z4aY2Wtmttr73nzOjzibM7NHzKzCzN5vZbmZ2Tzvc5WY2bhoxxirdIzzh45x/ojXYxzoOBcpof5fjAUtfQfMrJeZvWxmG72fPf2MsSVmNtj7f9V0TLjDa4+H2DPNbLmZvevF/j2vfaj3N3GT9zeyi9+xtsbMUr1j2t+813ERu5ltNbP3zGyNmRV7be3/zjjn4uoBpAIfAicDXYB3gZHN1rkV+K33/CrgSb/jbkfsU4Bu3vNvxFPs3npZwBvAO0CR33G34/c+HFgN9PRe9/U77nbE/hDwDe/5SGCr33F7sZwHjAPeb2X554DFgAFnA8v8jjkWHjrGxW7s3no6xkU/9pg8xnnx6Djnw3cilh4tfQeAHwEzveczgR/6HWcLcQ8AxnnPs4APvP9f8RC7Ad295+nAMu//10LgKq/9t03HjVh8AN8C/gz8zXsdF7EDW4Hezdra/Z2Jxx7QCcAm59xm59wR4Ang8mbrXA78wXv+NDDVzCyKMbamzdidc6855w55L98B8qIcY2tC+b0DfB/4IVATzeDaEErsXwMedM7tBXDOVUQ5xtaEErsDenjPs4EdUYyvVc65N4CqE6xyOfCYC3gHyDGzAdGJLqbpGOcPHeP8EbfHONBxLkJC/b8YE1r5DgQfo/8ATI9mTKFwzu10zq3ynh8A1gODiI/YnXPuoPcy3Xs44AICfxMhRmMHMLM84FLgf73XRpzE3op2f2fiMQEdBGwPel3mtbW4jnOuHqgGcqMS3YmFEnuwmwlcOY0FbcbuDS0a7Jx7IZqBhSCU3/upwKlm9qaZvWNmF0ctuhMLJfZ7gS+bWRmwCPiP6ITWae39/5AsdIzzh45x/kjkYxzoONcRifA76+ec2+k9/wjo52cwbfGmcYwl0JMYF7F7Q1jXABXAywR6zfd5fxMhtr83PwfuAhq917nET+wOeMnMVprZLV5bu78zaZGKTjrHzL4MFAHn+x1LKMwsBfgZcKPPoXRUGoEhapMJ9Mi8YWZnOOf2+RlUiK4GHnXO/dTMzgH+aGajnXONbW0o4hcd46JOxzgRHzjnnJnF7D0Pzaw78H/Afzrn9gcPponl2J1zDcAYM8sBngFO8zei0JjZ54EK59xKM5vsczgd8SnnXLmZ9QVeNrN/BS8M9TsTjz2g5cDgoNd5XluL65hZGoEhO3uiEt2JhRI7ZvYZYBYwzTlXG6XY2tJW7FnAaOB1M9tKYCz+czFSpCOU33sZ8Jxzrs45t4XAXIjhUYrvREKJ/WYCcwdwzr0NZAK9oxJd54T0/yEJ6RjnDx3j/JHIxzjQca4jEuF3tqtpqLX3M1aGvB/DzNIJJJ9/cs79xWuOi9ibeBfRXgPOITDEvalzLVa/N5OAad7fkScIDL39BfERO865cu9nBYHEfwId+M7EYwK6AhjuVYvqQqAAx3PN1nkOuMF7/gXgVefNjPVZm7Gb2VhgPoETs1j6T3/C2J1z1c653s65AudcAYG5XdOcc8X+hHuMUL4zfyXQM4CZ9SYwXG1zFGNsTSixlwJTAczsdAInZ5VRjbJjngOut4CzgeqgIRzJTMc4f+gY549EPsaBjnMdEcp3ItYFH6NvAJ71MZYWefMOHwbWO+d+FrQoHmLv4/V8YmZdgQsJzGF9jcDfRIjR2J1z33HO5Xl/R64i8Pf7WuIgdjM7ycyymp4DnwXepyPfmbaqFMXig0BVuQ8IjPee5bXNIXAyAIE/Tk8Bm4DlwMl+x9yO2F8BdgFrvMdzfsccauzN1n2dGKkQGeLv3QgMr1sHvIdXiSwWHiHEPhJ4k0ClwDXAZ/2O2YvrcWAnUEeg9+Vm4OvA14N+5w96n+u9WPq++P3QMS42Y2+2ro5x0Ys9Jo9xXmw6zkXpOxGrj1a+A7nAEmCjd8zr5XecLcT9KQLz+UqCjsefi5PYCwlU9S4hkADN9tpP9v4mbvL+Rmb4HWsbn2Myn1TBjfnYvRjf9R5rg47X7f7OmLehiIiIiIiISETF4xBcERERERERiUNKQEVERERERCQqlICKiIiIiIhIVCgBFRERERERkahQAioiIiIiIiJRoQRURERERETaZGZvtXP9yWb2t0jFI/FJCaiIiEgMsAD9XRaRmOWcO9fvGCT+6Q+dxCQzG29mJWaWaWYnmdlaMxvtd1wiIuFkZgVmtsHMHiNwQ/XBfsckItIaMzvo/ZxsZq+b2dNm9i8z+5OZmbfsYq9tFXBl0LYnmdkjZrbczFab2eVe+y/MbLb3/CIze0MX4xKbOef8jkGkRWY2F8gEugJlzrn7fQ5JRCSszKwA2Ayc65x7x+dwREROyMwOOue6m9lk4FlgFLADeBP4b6AY2AhcAGwCngS6Oec+b2b/A6xzzi0wsxxgOTAWcMAK4Dbgt8DnnHMfRvNzSXSl+R2AyAnMIXBAqgFu9zkWEZFI2abkU0Ti0HLnXBmAma0BCoCDwBbn3EavfQFwi7f+Z4FpZvZt73UmMMQ5t97Mvga8Adyp5DPxKQGVWJYLdAfSCRykPvY3HBGRiNCxTUTiUW3Q8wbazisM+Dfn3IYWlp0B7AEGhik2iWEaXy2xbD7w/4A/AT/0ORYRERERObF/AQVmdor3+uqgZS8C/xE0V3Ss9zMf+C8Cw3EvMbOJUYxXfKAEVGKSmV0P1Dnn/gz8ABhvZhf4HJaIiIiItMI5V0NgyO0LXhGiiqDF3ycwqq3EzNYC3/eS0YeBbzvndgA3A/9rZplRDl2iSEWIREREREREJCrUAyoiIiIiIiJRoQRUREREREREokIJqIiIiIiIiESFElARERERERGJCiWgIiIiIiIiEhVKQEVERERERCQqlICKiIiIiIhIVPz/kKKUdHaLYycAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 936x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "c_e_0 = model.initial_conditions[c_e].evaluate()\n",
    "c_s_0 = model.initial_conditions[c_s].evaluate()\n",
    "y0 = model.concatenated_initial_conditions.evaluate()\n",
    "\n",
    "fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(13,4))\n",
    "ax1.plot(x_fine, 1 + 0.1*np.sin(10*x_fine), x, c_e_0, \"o\")\n",
    "ax1.set_xlabel(\"x\")\n",
    "ax1.legend([\"1+0.1*sin(10*x)\", \"c_e_0\"], loc=\"best\")\n",
    "\n",
    "ax2.plot(x_fine, np.ones_like(r_fine), r, c_s_0, \"o\")\n",
    "ax2.set_xlabel(\"r\")\n",
    "ax2.legend([\"1\", \"c_s_0\"], loc=\"best\")\n",
    "\n",
    "ax3.plot(y0,\"*\")\n",
    "ax3.set_xlabel(\"index\")\n",
    "ax3.set_ylabel(\"y0\")\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The discretised rhs can be evaluated, for example at `0,y0`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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GzHnUkdF2XVoWzJznTTzt5A3MZPzwbF7cst/rUCQJOeeed841hRdXAf64MyMiIr718K2lLLh+EhPyB7Lg+kk8fGtpzGPYV1vHrIdfZt/hupgfW+KXElCJib1F1zKn4XYOZ4wEDHJGwzULPR+AqLV/Gfwq39pxM+7+XHhgkm9G6ZWk8ylgWWcvmtkdZrbWzNZWVanFXkREvKNReKU3NAquxMTfNlfxdPB87vzklzljxECvwzlR+WKuq/gmKRa+g1ezwzej9EpiMLPlwIgOXprrnHsqvM1coAl4rLP9OOceAR4BKC0t1ahZIiIScxqFV/pCCajExN+3VDF8YAbjh2d7HUrHVswnpbld+UjLKL1KQCUCnHOXdPW6mX0C+DAw02k4ZhER8bGVc2awYOkmnt+4h7rGIJlpAS6fOIK5V5/pdWgSB5SAStQ553j5nQNcePowzMzrcDrm81F6JbGZ2RXAHOBC59xRr+MRERHpikbhlb5QAipR99beIxx4r4HzTh3idSidyykIld12tF4k+h4EMoAXwjdpVjnnPuttSCIiIp1rGYX35qmFPP5KBVUaiEi6SQmoRN1Lb4dGlv3guKEeR9KFmfNCfT4bj72/zkej9Epic86N8zoGERGRnmg96u6C6yd5GInEG42CK1H3j3cOUDSkH6Nys7wOpXPFs+Cahbic0QQxDqYO990ovSIiIiIi8U4JqERVU3OQ1VsPcN6pPm79bFE8C7t7A1847c9cnfK/uLNu9DoiEREREZGEogRUomrDrloO1zfxQT/3/2zn3FOHsLumjh0Hj518YxERERER6TYloBJV/3gn1P/T1wMQtXPeKYMBWLX1gMeRiIiIiIgkFiWgElUvv3OAM0ZkM3RAhtehdNupwwYwdEC6ElARERERkQjzdQJqZleY2WYze9vM7u3g9Qwz+0349dVmVuRBmNKJ+qZm1mw7GFetnwBmxrRThrBq6wGcc16HIyIi4iu6PhOJrH21dcx6+GX2haeyab/c3XXR3CaSfJuAmlkK8BBwJTAB+JiZTWi32e3AofAUBg8A34ptlNKVsopq6hqDTI+HAYjaOfeUIexSP1AREZE2dH0mEnkLV2xhzbaDLFy+pcPl7q6L5jaRZH5t4TGz84D7nXOXh5e/DOCc+0arbZ4Lb/OymaUCe4BhrosPVVpa6tauXRvd4AWAB154ix/+eQvrvnIZAzPTvA6nR97ed5hLvv93vv2RYmZNGe11ONIFM3vVOVd68i0Tj85nIoklHs5nuj4TiZzx9y2jvinodRgnlZEaYPOCK3v0nq7OZ75tAQVGATtaLVeG13W4jXOuCagB4qveM4G98u5BJuQPjLvkE9QPVEREpBMRuz4zszvMbK2Zra2qqopSuCL+tXLODK6dnE9mWigly0g1RuVmkZEaWs5MC3D5xOFcPnH48W06Wted9/V2m+sm57PynhkR/dx+TkAjRie42GtoClK24xBTigZ7HUqvmBnTxqofqIiISLQ45x5xzpU650qHDRvmdThxIdp98yS28gZmkp2RSn1TkIzUAA3Njn7pKTQ0h5brm4IMG5DB0AEZx7fpaF133tfbbbIzUsnLzozo5/ZzAroTaF37WBBe1+E24RKPHOCEJiud4GKsfDHugUm8EbiJOW98FMoXex1Rr5x7ymD1AxUREWkrYtdn0nPR7psnsbf/SD23TBvDkjunc8u0MdQca2yzXHWk/oRtOlrXnff1dptI83Mf0FTgLWAmoRPZGuBm59zGVtt8HjjLOfdZM7sJ+Cfn3Kyu9qs+BlFWvhiemQ2NrZK2tCy4ZiEUd/mr8Z0tew9z6QPqB+p38dBnKlp0PhNJLPFwPtP1mTc66yvYm755IrEQl31Aw30G7gKeAzYBi51zG81svpldG97sZ8AQM3sb+DfghKHAJcZWzG+bfEJoecV8b+Lpg3F5AxjSX/1ARUREWuj6zBvt+wpGq2+eSCykeh1AV5xzS4Gl7dbNa/W8Drgx1nFJF2oqe7bex8yMqWMHs2b7Qa9DERER8Q1dn8Ve+76C0eqbJ9G3r7aOu35dxoM3lyTt78+3LaASp3IKerbe5z4wZhA7Dh5jb606+8dE+WIavzsBd38uPDApbvsPi4iIRFpH/QAl/qgfr89bQCUOzZxH05NfIDXYKmFLy4KZ8zp/j4+1jOK7dtshri4e6XE0CS7cfzitpYS7ZkeoPzHEXf9hERGRSHv41ve70y24fpKHkUhvtO/Hu2h1BYtWVyRlP161gEpkFc9iUd6/s5thOAxyRsflAEQtJuQPJCsthTXbVIYbdQnUf1hERCRZaaqYjqkf7/uUgEpEOef434Pn8N/jf4vdXw13b4jb5BMgLSXAnUNe485114HKQqMrgfoPi4iIJCuVmHZM/XjfpxJciahQf8l6po4d7HUokVG+mM/W/oC0lpJilYVGT05B6N+3o/UiIiLiayoxPbmWfrw3Ty3k8VcqqErSVmIloBJRq98NTVkytShBEtAV899PPlu0lIUqAY2smfNoWHIX6a7VoApx3H9YREQkmaycM4MFSzfx/MY91DUGyUwLcPnEEcy9+kyvQ/MN9eMNUQmuRNSabQfJyUrjtLwBXocSGSoLjZ3iWfx36uc4kJoHCdB/WEREJJmoxFS6Sy2gElFrtx2idMwgAgHzOpTIUFlozOypqePRw1MpuPo2Pv2hU7wOR0RERHpIJabSHUpAJWIOvdfA1v3v8ZEPJFByNnNeqM9n69FZVRYaFWu3h0YanpIo5dsiIiJJRiWm0h0qwZWIWVdZDUBJYa6ncURU8Sy4ZiFHs/IJOqNhwCiVhUbJq9sPkZkWYEL+QK9DEREREZEoUQIqEbOuopqAQXFBrtehRFbxLGo++xqn1D/GY+c9q+QzSsoqqikelUtaik5LIiIiIolKV3oSMWU7qjl9eDYDMhKvsntkThajcrNYu/2Q16EkpPqmZt7YVZtYreciIiIicgIloBIRwaBjXcWhhE4gSosGsXbbQZxzXoeScN7YVUtDc5DJo3O9DkVEREREokgJqETE1v3vUVvXRMnoQV6HEjWlYwaxt7aeykPHTr6x9EhZRTUAJYWJ+/0RERERESWgEiFlFaHS1ERuAW1Jjsp2VHsbSAIq21HNyJxMRuRorjARERGRRKYEVCJi3Y5qsjNSOXXYAK9DiZozRmSTmRY4nmxL5Kzbkdjl291lZv9uZs7Mhnodi4iIiEg0KAGViCirqGZyYS6BgHkdStSkpgQoLsjltXC5qERG1eF6dhw8lvT9P81sNHAZUOF1LCIiIiLRogRU+uxoQxNv7qlNigTinMJBvLGrhrrGZq9DSRjrwiXN6v/JA8AcQKNciYiISMLyZQJqZt8xszfNrNzMlphZbifbbTOz9Wa2zszWxjhMCSuvrCHoErv/Z4uSwlwamx0bd9V6HUrCWLfjEKkBY1J+jteheMbMrgN2Oude78a2d5jZWjNbW1VVFYPoRER0bSYikePLBBR4AZjknCsG3gK+3MW2M5xzk51zpbEJTdprGcF0cgKPgNuiJNzKq36gkVNWUc0ZI7PJSk/xOpSoMrPlZrahg8d1wH8C87qzH+fcI865Uudc6bBhw6IbtIjI+3RtJiIR4csE1Dn3vHOuKby4CijwMh7pRPlieGASn/3rB1iV9UUGv/Ok1xFFXd7ATEblZh1PuqVvmoOO13dUJ/T0PS2cc5c45ya1fwBbgbHA62a2jdD57jUzG+FlvCIirenaLPb21dYx6+GX2Xe4zutQRCLKlwloO58ClnXymgOeN7NXzeyOznagkrUoKF8Mz8yGmh0YjhGuKrRcvtjryKLunDGD1AIaIVv2Hea9huakKN/ujHNuvXMuzzlX5JwrAiqBc5xzezwOTUSkM32+NgNdn53MwhVbWLPtIAuXb/E6FJGISvXqwGa2HOjoDv9c59xT4W3mAk3AY53s5nzn3E4zywNeMLM3nXN/b7+Rc+4R4BGA0tJSDfARCSvmQ+Oxtusaj4XWF8/yJqYYKRmdyzOv72JPTZ3mreyjdeGWZA1AJCLivVhem4Guzzoz/r5l1DcFjy8vWl3BotUVZKQG2LzgSg8jE4kMzxJQ59wlXb1uZp8APgzMdM51eFJyzu0M/9xnZkuAqUCHJzmJsJrKnq1PIC2tdWUVh7jyrJHeBhPnyiqqye2XRtGQfl6H4hvhVlARkZjTtZk/rJwzgwVLN/H8xj3UNQbJTAtw+cQRzL36TK9DE4kIX5bgmtkVhKYjuNY5d7STbfqbWXbLc0Lz522IXZRJLqeTrh+drU8gE/NzSE8N8JrKcHsv3H/4GxsuYLl9Hlv/W68jEhGRLujaLHbyBmaSnZFKfVOQjNQA9U1BsjNSyctW1VU8Ul/eE/kyAQUeBLIJlW6sM7MfA5hZvpktDW8zHHjRzF4HXgGedc79yZtwk9DMeZCW1XZdWlZofYJLTw0wKX+gBiLqrVb9hwM4hjbvS5r+wyIicUzXZjG0/0g9t0wbw5I7p3PLtDFUHan3OiTpJfXlPZFnJbhdcc6N62T9LuCq8POtwNmxjEtaKZ5FU9Cxd8l/km8HsJyCUPKZ4P0/W5QUDmLRqu00NAVJT/XrfRyfSuL+wyIi8UrXZrH18K3vz2Cz4PpJHkYivaW+vJ3TlbP02sYhlzO9fiHLPrIJ7t6QVMnDOYWDqG8Ksml3rdehxJ8k7j8sIiIiyWHlnBlcOzmfzLRQupWZFuC6yfmsvGeGx5F5Twmo9FrLVCSTR+d6G4gHPnh0BS+mz6b4Z0XwwCSVj/ZEEvcfFhERSTbJ2gdSfXk7pwRUeq1sRzXDB2YwMtmmIilfTO6K/6AgsB/DQc0O9WHsiZnzcEnaf1hERCTZJHMfSPXl7Zgv+4BKfFi3o5qS0YMwM69Dia0V8zH1Yey94lkcONJA3Z++wqhA8vUfFhERSQbqA6m+vJ1RAgo0NjZSWVlJXV1ylQb0RXPQcd/0HHKyUtm0aZPX4bSRmZlJQUEBaWlp0TmA+jD22Uv9ZvDFhkE8O/t8JubneB2OiIiIRJjmM5XOKAEFKisryc7OpqioKPla83qp9lgjwQPvceqwAfTP8M/XyDnHgQMHqKysZOzYsdE5SE5BqOy2o/XSLWUV1WSlpTB+eLbXoYiIiEgUqA+kdKbbfUDNrJ+Z/ZeZ/SS8fJqZfTh6ocVOXV0dQ4YMUfLZA0cbmjGMrLQUr0Npw8wYMmRIdFuzk3gO1Egp21HNWQU5pKb4uxu6mX3RzAZayM/M7DUzu8zruEREROKB+kBKR3rSdPV/wKvAeeHlncBvgT9GOigvKPnsmaMNTWSmBQgE/PfvFvXfZUtfxRXzCdZUciBlGMOu+br6MHZTXWMzb+yq4VPnR6mFOrI+5Zz7gZldDgwCbgV+BTzvbVgiIiL+pz6Q0pGeND+c6pz7NtAI4Jw7Cvgv+5Coc85xrKGZfun+av2MqeJZcPcG7i95kQsbf0jzpBu9jihubNxVS2Ozo2T0IK9D6Y6Wc9xVwK+ccxvReU9ERESk13qSgDaYWRbgAMzsVEDt6BHyqU99iry8PCZN6tndoW984xuMGzeO8ePH89xzz3W4zYMPPsi4ceMwM/bv3398vXOObdu28eijj3Z5jF27dvHRj370+HJ9U5Bm5+iX3rYB3TnHxRdfTG1tbZef6eDBg1x66aWcdtppXHrppRw6dKhH8bR8pp///Ocn3S7aSgpzOdrQzFt7D3sdStxomT+2pDDX20C651Uze55QAvqcmWUDwZO8R0REREQ60ZME9CvAn4DRZvYYsAKYE5WoktAnPvEJ/vSnP3X6elFR0Qnr3njjDZ544gk2btzIn/70J+68806am5tP2G769OksX76cMWPGtFn/2c9+lhdffJGKigpuv/12du7c2eGx8/Pz+d3vfnd8+WhDE8AJLaBLly7l7LPPZuDAgV1+pm9+85vMnDmTLVu2MHPmTL75zW/2KB4IJbc//OEPO309ViaHW/HKKqq9DSSOrNtRTX5OJsMHxsUgBLcD9wJTwlUf6cAnW140s4leBSYiIiISj7qdgDrnXgD+CfgE8Gug1Dn315bXdSHWNxdccAGDBw/u0XueeuopbrrpJjIyMhg7dizjxo3jlVdeOWG7kpKSDhPYH/3oR/z617/m5z//Od/4xjcYNWoUf/vb35g8eTKTJ0+mpKSEw4cPs23btuOtmI8++ii33DSLO2/9KJMmnMGcOe/fg3jssce47rrrTvqZnnrqKW677TYAbrvtNp588slO4/niF7/I/PnzAXjuuee44IILCAaD9OvXj6Kiog4/bywVDelHbr801u045Gkc8aSsopqSwrgov8U5F3TOveacqw4vH3DOlbfa5FfeRCYiIiISn3o0f4Zz7gDwbCcv/wo4p88Reeyrz2zkjV21Ed3nhPyBfOWayOfnO3fu5Nxzzz2+XFBQ0GWrYXt33XUXH/vYx9i6dStz587lq1/9Kt/97nd56KGHmD59OkeOHCEz88RWqg3l5fzxLy8xPn8w48eP5wtf+AKjR4/mpZde4uGHHz7pcffu3cvIkSMBGDFiBHv37u00nm984xtMmTKFD33oQ8yePZulS5cSCITum5SWlrJy5UqmTp3a7c8caWZGyehctYB2077DdeysPsYnpxd5HUqkqD+oiIiISA9Ecg4EXYhF2Ne//vXjrZG7du06/vzzn/98RPb/ox/9iPPPP5/CwkJ+8pOfkJ+fz/Tp0/m3f/s3Fi5cSHV1Nampbe9RBIOOqdMvYMTQwWRmZjJhwgS2b98OhPp2Zmf3bF5HMzs+am1H8fTr14+f/OQnXHrppdx1112ceuqpx9+bl5fHrl27+viv0HclhYN4u+oItXWNXofie+vCifrk0bmexhFBzusAREREROJJj1pATyIhLsSi0VLZW3PnzmXu3LlAqA/ounXr2rw+atQoduzYcXy5srKSUaNGdXv/ZkZRURGf+MQnjq+79957ufrqq1m6dCnTp0/nueeea9MK2tAcJC09/Xj/z5SUFJqaQn1CU1NTCQaDx1soOzN8+HB2797NyJEj2b17N3l5eZ3GA7B+/XqGDBlyQrJZV1dHVla7+Tg9MHl0Ls5B+Y4azj9tqNfh+FrZjmpSA8akUTlehyIiIiIiHvD3LPDSpWuvvZYnnniC+vp63n33XbZs2dLnctR33nmHs846i3vuuYcpU6bw5ptvtnm9oSk0AGhW2olTsIwfP56tW7d2K+5f/OIXAPziF79o02+0ve3bt/O9732PsrIyli1bxurVq4+/9tZbb/V41OBoODvcmtcyuqt0rqziEBPyB5LZwfcnTjV4HYCIiIhIPIlkAqoLsT742Mc+xnnnncfmzZspKCjgZz/72UnfM3HiRGbNmsWECRO44ooreOihh0hJCV3YX3XVVcdbDBcuXEhBQQGVlZUUFxfz6U9/utN9/s///A+TJk2iuLiYtLQ0rrzyyjavNzQFSQkYqSknfnWuvvpq/vrXv570M91777288MILnHbaaSxfvpx77723w1icc9x+++1897vfJT8/n5/97Gd8+tOfpq6uDoCXXnqJSy+99KT/TtGWk5XGuLwBlO2o9joUX2sOOsorayiJo/JbM7vBzHJaLeea2fUty865czt8o4iIB8wsYGYDvY5DRKQr5lz3KmfN7Abgz865mvByLnCRc+7JqEUXBaWlpW7t2rVt1m3atIkzzzzTo4jih3OOTbsPk52ZyujB/U54fffu3Xz84x/nhRdeiHosZWVlfP/73+dXv+p4ENJY/06/9NvXWb5pL6/916XH+7RKW5t213LlD1bywP87mxtKCiK2XzN71TlXGrEdtt33Oufc5HbrypxzJdE4Xk91dD4TkfjVm/OZmT0OfBZoBtYAA4EfOOe+E4UQo0bnM5HE0tX5rEfzgLYknwDhaQm+0sfYOmRm95vZTjNbF35c1cl2V5jZZjN728w6bkaTiGlsDtIUDJ4w/2eLkSNH8pnPfIba2siOItyR/fv387WvfS3qx+muyYW5HDraSMXBo16H4lstIwWXjI6PKVjCOjpHRrLvvIhIX01wztUC1wPLgLHArdE4kK7PRCQSenIhFesLsQecc9/t7EUzSwEeAi4FKoE1Zva0c+6NKMaU1I42NAN0moACzJo1Kyax+KH0trWWpKqsopoxQ/p7HI0/lVUcYlC/NMYMObH13MfWmtn3CZ1rAD4PvOphPCIi7aWZWRqhBPRB51yjmUVzYEhdn4lIn/SkBXStmX3fzE4NP76PtxdiU4G3nXNbnXMNwBNA56PZSN8cPciAms2cZe+SeehNOHrQ64h85fThA+iXnqKBiLpQtqOayaNz461E+QuE+rf/htA5po5QEioi4hcPA9uA/sDfzWwMEP1SpM7p+qwX9tXWMevhl9l3uM7rUESiricJaKwvxO4ys3Iz+7mZdVSzNwrY0Wq5MrzuBGZ2h5mtNbO1VVVV0Yg1sR09CDU7SHVNmIE1N0LNDiWhraSmBDhrVA7rNBBRh2qONfL2viOUFMZV+S3Oufecc/c650qdc1Occ//pnHuv5XUz+6GX8YmIOOcWOudGOeeuciHbgRlRPKSuz6Jg4YotrNl2kIXLt3gdikjUdbuENnzR1Wkdv5n90Dn3he7uz8yWAyM6eGku8L/A1wjNLfo14HvAp7q77/acc48Aj0Cok3tv95O0Du8GF2y7zgVD6/sN9iYmHyopHMRPV26lrrE5kaYZiYjyymoASgpzPY0jCqZ7HYCIJDczywA+AhTR9rpufi/3p+uzGBp/3zLqm96/xlq0uoJFqyvISA2wecGVXbxTJH5Fsg9njy7EnHOXdGc7M/sJ8McOXtoJjG61XBBeJ5HW3MkMO52tT1Ilhbk0BR0bd9XwgTFKzI8rX8zZz/4XWzP2wDMFcMlXoDg2fYVFRJLAU0ANoW5R9X3dma7PYmvlnBksWLqJ5zfuoa4xSGZagMsnjmDu1ZqdIR7tq63jrl+X8eDNJeRlZ3odjm/5cjRHMxvpnNsdXrwB2NDBZmuA08xsLKET203AzTEKMbmkpHecbKakxz4WH2uZ37KsoloJaIvyxfDMbAY2HgMDaivhmdmh15SEiohEQoFz7opYHEjXZ5GXNzCT7IxU6puCZKQGqG8Kkp2RquQlTrUupV5ww1leh+NbPekDGkvfNrP1ZlZOqB/D3QBmlm9mSwGcc03AXcBzwCZgsXNuY0yiK18MD0yC+3NDP8sXx+SwPeGcY/bs2YwbN47i4mJee+213u8seyRB2g0cYwHIHtm3IBNM3sBMRuVmUaZ+oO9bMR8aj7Vd13gstD4xxNWISiKSkP5hZrG60vX39Vmc2n+knlumjWHJndO5ZdoYqo70uSFbYmz8fcsouvdZFq2uwLlQKXXRvc8y/r5lXofmS5FsAY3YhZhzrsP5q5xzu4CrWi0vBZZG6rjdEm7ROX5RXbPDly06y5YtY8uWLWzZsoXVq1fzuc99jtWrV/duZ/0Gs6f6GHkcJJWmUMtn9kj1/+zA5MJc1oXnuxSgprJn633MzALAgPB8ey1+4FU8IpLczGw9oWuvLOCTZraVUAmuAc45VxzpY/r6+iyOPXxr6fHnC66f5GEk0lsqpe6ZXrWAmlnAzAa2W50cF2JRatH55S9/SXFxMWeffTa33trx/NFVVVV85CMfYcqUKUyZMoWXXnqp0/099dRTfPzjH8fMOPfcc6murmb37t2dbt+VxuYg+4P9OZR9OuSXwPCJSj47UTI6l53Vx9hXq2HUAcgp6Nl6nzGzx81soJn1J1Rq9oaZfanldefcoxE81hfM7E0z22hm347UfkUkYX0YuAYYDowDLgsvt6wXkRhRKXXPdDsBjeWFmK9FoUVn48aNLFiwgD//+c+8/vrr/OAHHefyX/ziF7n77rtZs2YNv//97/n0pz/d6T537tzJ6NHvjwFQUFDAzp29GwPgaEMzAP3SNbLryVzc+FdeTJ/NsO+P8G15dkzNnEeDZbRdl5YFM+d5E0/PTQi3eF4PLAPGAh3fIeoDM5tBaJ68s51zE4FOJ3kXEQFwzm13zr0L/B7ICy8ff3gdn0hXEnHeU5VSd19PSnAnOOdqzewWQhdi9xIace07UYnMr3IKQmW3Ha3vpT//+c/ceOONDB06FIDBgztuXVy+fDlvvPHG8eXa2lqOHDnCgAEDen3s7jja0ISZkaWpRbpWvpix//hPLODv8uyYKp7Fwuc3c9uxXzKsuSr0/2TmvHj690gzszRCCeiDzrlGM4vGVAGfA77pnKsHcM7ti8IxRCQxTQNuMbPtwHtEsQRXJFIScbAelVJ3X08S0FhdiPnbzHlt+4BCzFp0gsEgq1atIjPz5M35o0aNYseO9xPlyspKRo3qcB7okzra0ExWWoBAQOOtdGnFfKypk/Ls+Em4IqqusZkfH/wAzRfcyD1XnOF1OL3xMLANeB34u5mNAWq7fEfvnA58yMy+DtQB/+GcW9PRhmZ2B3AHQGFhYRRCEZE4c7nXAYh0l+Y9FehZH9CWC7H+RPdCzN+KZ8E1CyFnNGChn9cs7FOCcfHFF/Pb3/6WAwcOAHDw4MEOt7vsssv44Q9/eHx53bp1ne7z2muv5Ze//CXOOVatWkVOTg4jR/Z81FrnHMcamumX7ssZe/wlgQbciZSNu2poCjomh6eoiTfOuYXOuVHOuatcyHZCIz/2mJktN7MNHTyuI3QzcDBwLvAlYLGZdXjHxzn3iHOu1DlXOmzYsN5+NBFJEO1Lb1WCK362cs4Mrp2cT2ZaKAXJTAtw3eR8Vt7Tqz+tEqe6nVU45xYCC1ut2h7ut5R4jh6Ew7tDc192NOJr8ayItmhNnDiRuXPncuGFF5KSkkJJSQmPPvroCdstXLiQz3/+8xQXF9PU1MQFF1zAj3/84w73edVVV7F06VLGjRtHv379+L//+79exVbXGCTonPp/dkcUyrPjXVl4ROCSOE1AzSwD+AhQRNvzZY9HHetqcncz+xzwB+ecA14xsyAwFKjq6XFERBLZvto67vp1GQ/eXKIBXuKQBusR6EECGskLMV87ejCURLhweUBzw/tJRRRHfr3tttu47bbbutxm6NCh/OY3v+nW/syMhx56qM9xHW1oAiBLCejJeVie7VdlFdWMys0ib2Dc/mF5Cqgh1N89mqMJPEmoZfUvZnY6kA7sj+LxRETiUiL2HUw2LYP13Dy1kMdfqaAqgQYiku7pSV1lrC7EvHV49/vJZwsXDK1PwqlHjjY0kxoIkJ7Sqxl7kku4VdytmI+rqaQmLY9B1yxI2v6fAGUVhzhnzCCvw+iLAufcFTE4zs+Bn5vZBqABuC3cGioiIqjvYCLRYD3SkwQ0Vhdi3mpu6Nn6KPj617/Ob3/72zbrbrzxRubOnXvCtv/3f/93wrQt06dPj0jrJ4QS0H7pKXTSHU3aK56FFc/i04+uYfvBoywvvtDriDyzt7aOXTV1fCpOy2/D/mFmZznn1kfzIM65BuCfo3kMEZF4tnLODBYs3cTzG/dQ1xgkMy3A5RNHMPfqM70OTUR6qCcJaEwuxLzinAslWSnpHSebKekxi2Xu3LkdJpsd+eQnP8knP/nJqMTRFAxS39RMbr+0qOw/WvzQcDR5dC4r3txHzbFGcrLi698vUsoqDgFQUhh/LaBmtp7QVAZZwCfNbCuhyg9NbyAi4gH1HRRJHCdNQJPhQiwzM5MDBw4wZMgQLHtk2z6gABYIDUSUZI41NAPQP476fzrnOHDgQLemqommlqSrvLKaD52WnCOVllVUk54SYNKogV6H0hsfJjRK+HpgnMexiIgI6jsokii60wKa8BdiBQUFVFZWUlUVHnCyoRHqanDBJoKWQkpWLlTvBfZ6GWbM1dY1cvhYEym1mQTiqAQ3MzOTggJvR54tHp2DWSgJS+YEdEL+QDJS4+cGRouWKQzM7PdAXmdzcoqISOyo76BIYjhpApoMF2JpaWmMHTv2hPWzf13GK+8eZNV/lnbwrsR3289fYW9tHX/613O8DiXuDMxMY9ywAcfLUJNNY3OQ8p3V3Dx1jNeh9NU04BYz2w68RwJVfoiIiIh4oSd9QJPuQuycwlyefn0Xu2uOMTIny+twYioYdJRVHOLq4uQrPY6UksJcXnhj7/v9i5PIm7sPU9cYpKQw1+tQ+upyrwMQERERSSQ9SUCT7kKspR/fa9urubo4uRLQrfvfo7auKS4HkPGLksJBLF5byfYDRyka2t/rcGKqbEfLAES53gbSRy0VICIiIiISGd2e3NE5t72jRzSD89qZIweSkRpIyjLKls98TpwnEF5qSb5akrFk8tr2Q+RlZzAqN7lu3IiIiIhI17qdgCaj9NQAZ43K4bUkTEBfq6hmYGYqpwwd4HUoceu0vGz6p6dQVlHtdSgxV7ajmpLC3KQrPRYRERGRrikBPYmSwlw27KqlvqnZ61BiqqziEJMLBxEIKIHorZSAUVyQy7od1V6HElMHjtSz/cBRlW+LiIiIyAl8mYCa2W/MbF34sc3M1nWy3TYzWx/ebm00YjmncBANTUE27T4cjd370pH6Jt7ae5iS0blehxL3SgpzeWNXLXWNyXMDoyXhPkcJqIhIwvDTtZmIxLeeDEIUM865/9fy3My+B9R0sfkM59z+aMXy/kBEh5icDAlZ+WJSn/sKb6fvoqEsH4Z/FYpneR1V3Jo8OpemoGPDzhpKiwZ7HU5MvFZxiJSAcdaoHK9DERGRCPHTtVki2Fdbx12/LuPBm0vIy870OhyRmPJlC2gLC3UgmwX82qsYRuRkkp+TSVkylFGWL4ZnZpP53i4CBplHd8Ezs0PrpVcmhwciSqYy3LKKas4cmU1WeorXoYiISIT54dosESxcsYU12w6ycPkWr0MRiTlftoC28iFgr3Ous/+dDnjezBzwsHPukY42MrM7gDsACgsLexxESeEgXtueBAMRrZgPjcfarms8FlqvVtBeycvOpGBQVtIMRNQcdLy+o5qPfKDA61BERCQ6InJtBn2/PotH4+9bRn1T8PjyotUVLFpdQUZqgM0LrvQwMpHY8awF1MyWm9mGDh7XtdrsY3R9h+1859w5wJXA583sgo42cs494pwrdc6VDhs2rMexlhTmsrP6GPtq63r83rhSU9mz9dItJYWDkmYqn7f2Hua9hua4n/9TRCQZxfLaDPp+fRaPVs6ZwbWT88lMC12CZ6YFuG5yPivvmeFxZCKx41kC6py7xDk3qYPHUwBmlgr8E/CbLvaxM/xzH7AEmBqNWI/3A030VqycTlqtOlsv3TJ5dC67aurYm+g3MMoXU/iLqWzNuJlr/nyZSrdFROJMPF2bxau8gZlkZ6RS3xQkIzVAfVOQ7IxU9QONU/tq65j18MvsO5zg13gR5uc+oJcAbzrnOmx+M7P+Zpbd8hy4DNgQjUAmjRpIekqAsh0J3oo1cx5NKe1OgGlZMHOeN/EkiJbWwIQuww33H+5ft5uAQerhneo/LCKSeHxzbRbP9h+p55ZpY1hy53RumTaGqiP1XockvaS+vL3j5z6gN9GuxMPM8oGfOueuAoYDS8IT3acCjzvn/hSNQDJSU5iQP5Cy7dXR2L1/FM9iyauVTN/2ECPtAJZTEEo+1f+zTybmv38D44pJI7wOJzrUf1hEJBn45tosnj18a+nx5wuun+RhJNJb6svbN75NQJ1zn+hg3S7gqvDzrcDZsYrnnMJBPP7Kdhqbg6Sl+LnhuG9+cWQKT45+jMc+fa7XoSSMjNQUzswfyLpEbgFV/2ERkYTnt2szEa+snDODBUs38fzGPdQ1BslMC3D5xBHMvfpMr0OLC4mbSUVYSWEudY1BNu857HUoUXOsoZlNuw9TMnqQ16EknI/3W833d/0z7v5ceGBS4pWmqv+wiIiI9FK89aVUX96+UQLaTeeMaRmIKHH7gZZXVtMcdBrBNNLKF3Nd5bcYZfsxHNTsSLz+kTPn0RhQ/2ERERHpuXjsS6m+vL3n2xJcv8nPySQvO4Oyimo+fp7X0UTHq+HkumXUX4mQFfNJbW53Ry/R+kcWz+KRv7zNR2t+zvDg/lDLp/oPi4iISBfiuS+l+vL2nlpAu8nM+JdBr3LPmx+FBC2jXLvtEKcO68/g/uleh5JYkqB/ZFNzkB8dOIeHzn4S7q+Guzco+RQRkaiLt9JNaUvzoiYnJaDdVb6Yj+//PiNcFSRgGWUw6Hh1+yGmFA32OpTEkwT9I9/cc5j3Gpr5wBi1nouISOzEY+mmvE99KZOTSnC7a8V80oKJW0b5dtURao41UqoENPJmzgvdrGg9TUmC9Y9cu+0ggG5giIhITMRz6aa01dKX8uaphTz+SgVVas1OeEpAuyvByyjXbgv1/yxVC1bkhW9Q1D33FdKP7Ka+/0iyrvhqQty4aLFm+yFG5WaRn5vldSgiIpIENA1G4lBfyuSjEtzuSvAyyrXbDjJ0QAZjhvTzOpTEVDwL98UNjG96nIXFSxIq+XTOsXbbQZXfiohIzKh0UyR+KQHtrpnzQmWTrSVQGeWa7QeZUjQIM/M6lISVlZ7CpFE5rHn3oNehRFTloWPsra1nSpESUBERiR1NgyESn1SC210tZZR/+grp7+2mof9IMhOkjHJvbR07Dh7jtvOKvA4l4U0pGsyjL22jrrGZzLQUr8OJiLXbQwn1B8ao/6eIiMSOSjdF4pNaQHuieBaNs8sZ1/AYPyp5KiGST3i//6cGkIm+KUWDaWgOUl5Z43UoEbN22yGyM1IZPyLb61BERERExOeUgPZQdmYaE/IHHh/1MxGs2XaQrLQUJuQP9DqUhNcyyNOaBPr+rN12iJIxg0gJqHy7t8xsspmtMrN1ZrbWzKZ6HZOIiIhINCgB7YXSMYMpq6imsTl48o3jwKvbDzF5dC5pKfo6RNug/umcljcgYRLQmqONvLXvMFM0AFFffRv4qnNuMjAvvCwiIiKScJRx9MKUosEca2zmjV21XofSZ0fqm9i4q0YDyMTQlLGDeXXbIZqDzutQ+uy1ikM4h+aP7TsHtJQg5AC7PIxFREREJGqUgPZCaVHilFGuq6gm6OADSiBiZmrRYA7XN/Hmnvi/gbFm20FSA8bk0blehxLv/hX4jpntAL4LfNnbcERERESiQwloLwwfmEnh4H4JkYCu3X6QgME5hbleh5I0jt/ASIDpWNZuP8TEUTlkpSfGiL7RZGbLzWxDB4/rgM8BdzvnRgN3Az/rYj93hPuJrq2qqopV+CIiIiIRoQS0l6YUDWbttkM4F99llGu2HWT8iIFkZ6Z5HUrSKBjUj/ycTNZsP+R1KH1S19jM6zuq1f+zm5xzlzjnJnXweAq4DfhDeNPfAp0OQuSce8Q5V+qcKx02bFgsQhcRERGJGM8SUDO70cw2mlnQzErbvfZlM3vbzDab2eWdvH+sma0Ob/cbM0uPTeQhU4oGceC9Bt7d/14sDxs55YtxD0zkVzuu4Nfv3Q7li72OKKlMGTuYNe8ejOsbGK/vqKa+Kci5pwzxOpREsAu4MPz8YmCLh7GIiEgE7autY9bDL7PvcJ3XoYj4gpctoBuAfwL+3nqlmU0AbgImAlcAPzKzjur7vgU84JwbBxwCbo9uuG21DLoSl2W45YvhmdlYTSUBc+Q27IVnZisJjaHSosHsO1xPxcGjXofSa6u2HsQslExLn30G+J6ZvQ78N3CHx/GISJKK9wYCP1q4Ygtrth1k4XLdWxQBDxNQ59wm59zmDl66DnjCOVfvnHsXeJt25WhmZoRaCX4XXvUL4PoohnuCU4f1Z+iAdFZtjcMEdMV8aDzWdl3jsdB6iYmp4RsYr8RxP9BVWw8wMX8gOVkq3+4r59yLzrkPOOfOds5Nc8696nVMIpK04rqBwE/G37eMonufZdHqCpyDRasrKLr3Wcbft8zr0EQ85cc+oKOAHa2WK8PrWhsCVDvnmrrYJqrMjGmnDOHldw7EXxllTWXP1kvEnZY3gEH90lgdjwlo+WKC35/IYzuvYNHhT6vlXEQkgcR7A4GfrJwzg2sn55OZFrrczkwLcN3kfFbeM8PjyKQ3VEodOVFNQE8y6mPMRGvUyPNOGcKe2jq2HYizMsqcgp6tl4gLBIxz4/EGRrh8O1DbUr69R+XbIiLJIeINBIk+qnfewEyyM1KpbwqSkRqgvilIdkYqedmZXocmvaBS6shJjebOnXOX9OJtO4HRrZYLwutaOwDkmllq+CTX0Tat43gEeASgtLQ0Ylf7550aGnzl5XcOMHZo/0jtNvpmzqP5qdmkNLcqw03LgpnzvIspCZ136hCWbdjDjoPHKBzSz+twuqer8u3iWd7EJCIiPWJmy4ERHbw0Nzwyd0xE6/rMT/YfqeeWaWO4eWohj79SQZVaz+LO+PuWUd8UPL68aHUFi1ZXkJEaYPOCKz2MLH5FNQHtpaeBx83s+0A+cBrwSusNnHPOzP4CfBR4gtAUBjE7YbY4ZWh/8rIzeHnrAW6eVhjrw/de8SyWb9zDxDd/wCg7gOUUhJJPJRAxdVnz37k4/auM+uGBUOtzPPwOVL4tIhL3/NJAkAwevvX9cZwWXD/Jw0ikt1bOmcGCpZt4fuMe6hqDZKYFuHziCOZefabXocUtL6dhucHMKoHzgGfN7DkA59xGYDHwBvAn4PPOuebwe5aaWX54F/cA/2ZmbxMq+eh04vYofgbOOzUOyyiBX703jc8MfhS7vxru3uD/xCfRlC9m+N/mUBDYj+GgZkd8lLKqfFtEJFk9DdxkZhlmNpZOGgiAlgYC8KiBQBKDX/pcqpQ68rwcBXeJc67AOZfhnBvunLu81Wtfd86d6pwb75xb1mr9Vc65XeHnW51zU51z45xzNzrn6r34HOedMoT9R+p5p+qIF4fvlYamIGu3H+TcUzR9hmdWzMficSTimfNoTslqu07l2yIiCSMRGggkMfipz2VLKfWSO6dzy7QxVB3xJO1IGH4swY0rM5v+xovp9zPqR/FTRlleWU1dY5BzTxnidSjJK15LWYtn8cLGPUxS+baISEJyzi0BlnTy2teBr3ew/qpWz7fSbnRckZ7wY59LlVJHlh+nYYkf5YsZ+pcvxV0Z5aqtBzCDaWPVAuqZOC5l/eWRqfzLEJVvi4iIP/mldFN6R9PXJD4loH0Rp2WUL769nzNHDCS3X7rXoSSvmfNCpautxUEp67GGZtZuP8QHT1XruYiI+JOfSjel59TnMvGpBLcv4rCM8mhDE69uP8Snpo/1OpTk1tJquGI+rqaS3QxhxIe/QcDnrYmvbDtIQ1OQ808b5nUoIiIibfixdFN6R9PX+M++2jru+nUZD95c0uebAUpA+yKnIFR229F6n1r97kEamx0fUgLhveJZUDyL379ayX/89nWW5X0Ivw/o/eKWKtJTAkwtUvm2iIj4i6bLSBzqc+k/rSsLFtxwVp/2pQS0L2bOC/X5bF2G6/MyypVv7ScjNUBp0SCvQ5GwlnLWl97ez5kjB3ocTddWbtlPadEgstJTvA5FRESkDZVuikReNCoL1Ae0L4pnwTULIWc0DmOnG0rDVf/j60FZXny7iqljB5OZpgTCL/JzsxiXN4C/vVXldShd2ne4jjf3HOb804Z6HYqIiEiHNF2GSGRFY1AotYD2VbiM8q9v7uOTj67hVwOm8iGvY+rE3to63tp7hI9+wL8lwsnqwtOH8atV2znW0Ozb1sWX3t4PwAUq3xYREZ9S6aZIZEWjskAtoBEy7ZTBpKcE+LuPW7FWbgklEOePUwLhNxeePoyGpiCr3j3gdSidWrllP4P7pzPB52XCIiIiIhI5ka4sUAtohPRLT2XK2EH8/a39zL3a62g69uKWKoYOSOeMEdlehyLtTB07mIzUAH/bXMWM8Xleh3MC5xwvbtnPB08dQiBgXocjIiIiIjES6coCtYBG0AWnDWPz3sPsrjl28o1jzDnHi28f4PxxQ5VA+FBmWgrnnjLEty3oW/YdYd/hej6k/p8iIiIi0gdKQCPogtNDpa0r39rvcSQn2rirlv1H6jX9io9dePowtu5/jx0Hj3odygn+8uY+4P3vuIiIiIhIbygBjaAzRmSTl53B37b4qBWrfDE8MImJPxnDixmzuaz5715HJJ24cHwoufPjaLgr3tzHmSMHMjIny+tQREREfG1fbR2zHn6ZfYfrvA5FxJeUgEaQmfGh04bx4pb9NAed1+GEks9nZkPNDgxHge0n+4V/C60X3zllaH8KBmX5LgGtOdrIq9sPMfMM//VNFRER8ZuFK7awZttBFi7f4nUoIr6kQYgibMYZw/j9a5W8VnGIKUWDvQ1mxXxobNcftfFYaL2P5ypNVmbGBacP46mynTQ0BUlP9fj+UPliWDGfgTWV/C11CI3p9wHjvY1JRETEp8bft4z6puDx5UWrK1i0uoKM1ACbF1zpYWQi/qIW0Ai74PRhpKUYy9/Y63UoUFPZs/XiuYvH5/FeQzOrtno8HUv71vPAfor+8WW1nouIiHRi5ZwZXDs5n8y00OV1ZlqA6ybns/KeGR5HJr2hUuroUQIaYQMz0zj3lCG84IcENKegZ+vFc+efNpSstBTvvz8dtJ5bS+u5iIiInCBvYCbZGanUNwXJSA1Q3xQkOyOVvOxMr0OTXlApdfSoBDcKLp0wnHlPbeSdqiOcOmyAd4HMnId7ZnYocWiRlgUz53kXk3QpMy2Ffx+5jqvX3YVbtx/LKQj9vmJdMq3WcxERkR7bf6SeW6aN4eaphTz+SgVVaj2LOyqljj7PWkDN7EYz22hmQTMrbbX+UjN71czWh39e3Mn77zeznWa2Lvy4KnbRd23mmcMBvG/FKp7F5ilfpzI4FIdBzmi4ZqH6f/pZ+WI+ceABRlKF4aBmR6gUNtalr2o9FxER6bGHby1lwfWTmJA/kAXXT+LhW0tP/ibplliVxKqUOvq8LMHdAPwT0H5ekP3ANc65s4DbgF91sY8HnHOTw4+lUYqzx0blZjExf6Av+oH+8r2pXMZD1M89AHdvUPLpdyvmk9rc7sTqRenrzHm4tHZTrqj1XEQk4SVyA4HEt1iVxKqUOvo8K8F1zm2C0Mif7daXtVrcCGSZWYZzrj6G4fXZ54e8RvFbC3H3H/CsjLI56Hh+4x4uPiOPzLSUmB5beskvpa/Fs9hVfQy3fD6jAt59h0VEJOZaGggebre+pYFgl5lNAp4DRnWyjwecc9+NYow9sq+2jrt+XcaDN5coiYhDXpTEqpQ6uvzeB/QjwGtdJJ93mdnHgbXAvzvnDnW0kZndAdwBUFhYGJVA2yhfzBXvfoOAhftetpRRQkwv4NdsO8j+Iw1cOWlkzI4pfZRTEPq+dLQ+xn5Tdy4PNi7klbmXMHRARsyPLyIisZeIDQStW84W3HCW1+FID62cM4MFSzfx/MY91DUGyUwLcPnEEcy9+syoHbN16fSC6ydF7TjJKqoluGa23Mw2dPC4rhvvnQh8C/iXTjb5X+BUYDKwG/heZ/tyzj3inCt1zpUOGzas5x+kp1bMJ9DUyfybMbRs/W4y0wJcND4Gn1kiY+a8UKlrax6Vvi7bsIepYwcr+RQRkfa600BQbmY/N7NBne3EzO4ws7VmtraqqiriQY6/bxlF9z7LotUVOBdqOSu691nG37cs4seS6FFJbOKJagLqnLvEOTepg8dTXb3PzAqAJcDHnXPvdLLvvc65ZudcEPgJMDXyn6CXfFBGGQw6/rRxDxeePoz+GX5v6JbjimeFBorKGU0QY29gmCcDR7297whb9h1R67mISAJKlgYCDSaTOFpKYpfcOZ1bpo2h6ojvG96lC77LTMwsF3gWuNc591IX2410zu0OL95AqM+CP/igjPK1ikPsra1XAhGPimdB8SweffFd5v/xDZaPuIBxMQ5h6frQf63LJ46I8ZETm5ndCNwPnAlMdc6tbfXal4HbgWZgtnPuOU+CFJGE55y7pDfv624DQavtfwL8sVdBRoBazhKHSmITi5fTsNxgZpXAecCzZtZysXUXMA6Y12oEtbzwe37aakS2b4dHYisHZgB3x/ozdMoHZZRPrttJZlqAmWfmxeyYElkfLh6JGTz9+u6TbxxBzjmeLNvJtLGDGZGjP9IR1uHo32Y2AbgJmAhcAfzIzDRymIj4Rk8aCFotet5AoJYzEf/xchTcJYTuorVfvwBY0Ml7Pt3q+a3Ri66PWsolV8zH1VSyMziE/jO/yqAYlVE2NAX5Y/luLpswguzMtJgcUyIvb2Am544dwh9f38Xdl5x2woAQ0fJ6ZQ1b97/Hv1x4SkyOl0w6G9wDuA54Ityf6l0ze5tQt4KXYxuhiCQ7M7sB+CEwjFADwTrn3OW0bSBouaN+mXNun5n9FPhxuKrj22Y2GXDANjov1Y0JtZyJ+I/vSnATRriMcvv+97jou3/ly/VnxOwM/JfN+6g+2sgN53Q2OrrEi2vOzuc/l6xnw85azirIickxl7xWSXpqgCtUvh1Lo4BVrZYr6WR6g5iP6i0iSSWhGwhExBc8K8FNFkVD+1NSmMtvX63EOReTY/7htUqGDsjgQ+OGxuR4Ej1XnTWC9NQAv3u1gz7FUdDYHOSZ8t1ceuZwcrLUet4bfRncoztiPqq3iIiISAQpAY2Bm6aM5u19R3itojq6BypfTPP3J/K/b89keeDzpG78XXSPJ1GX2y+dKyaOYEnZTuoam6N+vL+/VcXB9xq4oUSt573Vy9G/dwKjWy0XhNeJiIiP7autY9bDL7PvcJ3XoYjEDSWgMfDh4nz6p6fwmzUV0TtI+WJ4ZjYptZUEDHIb9sAzs0PrJa7dNGU0tXVN/GnDnugdpHwxPDCJi38znn9kzuaihr9G71jSkaeBm8wsw8zGAqcBr3gck4iInMTCFVtYs+0gC5dv8ToUkbihBDQG+mek8uHifP5Yvpsj9U3ROciK+dB4rO26xmOh9RLXzj1lCIWD+/FEtG5ghG9eULMDw5HPflKf/aJuXkRBZ6N/O+c2AouBN4A/AZ93zkW/yVtERHpl/H3LKLr3WRatrsA5WLS6gqJ7n2X8fcu8Dk16QS3ZsaUENEb+39TRHG1o5pnXd0XnADWVPVsvcSMQMP7flNGs2nqQd/e/F/kD6OZFzDjnljjnCpxzGc654eGRJVte+7pz7lTn3HjnnK5gRER8bOWcGVw7OZ/MtNCldGZagOsm57PynhkeRya9oZbs2FICGiMlo3M5c+RAHn1pW3QGI8op6Nl6iSs3fqCAG1JfYtDDJXB/LjwwKXItlLp5ISIi0iN5AzPJzkilvilIRmqA+qYg2Rmp5GVr/uxY6mvLpVqyvaEENEbMjNvPH8v4qmXUf2dCxJOIQ+fdy1GX3nZlWhbMnNfxGySu5G17mm+l/ZTcxr2Ag5odkevjq5sXIiIiPbb/SD23TBvDkjunc8u0MVQdqfc6pKTT15ZLtWR7Q/OAxtD1KS9xdfrPyDwaPkG1JBEQmje0Dx7afw77mz/Ddwc9RerhnaHkYea8Pu9XfGLFfNJduz9sLWWyffwdu5nzqP/DXWTSav+6eSEiIglsX20dd/26jAdvLul1q+XDt5Yef77g+kmRCk26Yfx9y6hvCh5fXrS6gkWrK8hIDbB5wZXd3o9asr2hFtAYSv3L18iikySiD6qPNvD4KxUw6UZS/30j3F8Nd29Q8plIolgm+7eMi5jTcDvvZY0EDHJGwzUL9f0REZGEpT5/8S2SLZdqyY49tYDGUpSSiEf/sY2jDc187qJxfdqP+FhOQajFvKP1feCc40d/eYeKAZeQ9u9fh1TdkxIRkcQVqZYz8VYkWy7Vkh17utqMpSj0tTtwpJ6frnyXyyYMZ/yI7F7vR3xu5rxQWWwrLrXvZbJ/fauKV7Yd5HMXnUq6kk8REUlw6vOXONRyGb/UAhpLM+eF+ny2mvLCpWVhfUgifvjntznW2MycK86IRITiVy3lsCvm42oq2Rkcwpun/SuX9KFMtjno+NayNxkzpB8fm1oYoUBFRET8S33+EodaLuOXEtBY6iCJOFhyL8U9TSLKFx/fx2fcEM489U7G5V0V+XjFX4pnQfEsDJj36BpWbzzAipo6RuT07o/m717dwZt7DvPDj5Wo9VNERJJGS8vZzVMLefyVCqp6MIVHJAYvEkl2uuqMteJZcPcGmv7rILcP+j8+V34qRxuauv/+8sWhVtSaHRiOUbafWbu/E7k5ISUufOWaCTQFHV/74xs9e2P5YnhgEu7+XD707Azuzivj6rNGRidIERERH3r41lIWXD+JCfkDWXD9pDYtaSejwYviX1/nDpW+UwLqkbSUAAtumMTO6mN8Y+mb3X/jivltSngBrKnvI+lKfBkzpD93zRjHs+t3s+7ZR0Jzyp5sbtl2Ny/y2c8Xjj5IYMNvYxq7iIhIvBl/3zKK7n2WRasrcC40eFHRvc8y/r5lXocmrXQnudRNBO8pAfXQlKLBfPr8sfxq1XZeX9rNJCKK03FIfPmXC0/lC0NfY/yaueERct37c8t29P3p4OZFQDcvREREgK6TFw1eFB+6Si51E8E/1AfUY3OuOIOsN3/P6asXgjWEVrYkERWrYMvzoeQypwBmzqMpexSphztINvs4HYfEn/TUAF+0J0iloe0LLXPLtu9brJsXIiIinWqdvCy44aw2r2nwIn/rzvQ6K+fMYMHSTTy/cQ91jUEy0wJcPnEEc68+06uwk5ZnLaBmdqOZbTSzoJmVtlpfZGbHzGxd+PHjTt4/2MxeMLMt4Z+DYhd95KSnBvhXe4Is6yCJWPvzNi1bwadn88zRszjm0ttum9b36TgkPqUe3tnxCzU7TmhRr+vXSV9P3bwQEYm4J8t2Mv2bf2bsvc8y/Zt/5smyTs7XPpOM12edtYydPndpmxZRTfvhX121ULe0bGPoJoJPeFmCuwH4J+DvHbz2jnNucvjx2U7efy+wwjl3GrAivByXUjpLInBtlgJNx5jWvIbdF34bckYDFvp5zcITW7skOXSSPDqszc2Lhie/wOLaCdSR0XZD3bwQEYm4J8t28uU/rGdn9TEcsLP6GF/+w/p4SUKT7vqss+TlmrPz25Rz9mXwIomurlqoW7ds6yaCP3hWguuc2wRgZr3dxXXAReHnvwD+CtzT17g8kVMQThZObiQHsIs/CRd/MspBSVzoYG7ZoIOAtb15kR6s45qs9diVC+FvC9qUdevmhYhIZH3nuc0ca2xus+5YYzPfeW4z15eM8iiq7knG67P2yUtdY5Cn1u06/npH5ZziP+2n13l89XYWra44/nrL84zUAAuun6S5Qz3k1z6gY82sDKgF7nPOrexgm+HOud3h53uA4Z3tzMzuAO4AKCwsjHSsfddBEuGAjk79pnJJaa3V3LLUVOJyCrBObmYMatwH59wUeoiISNTsqj7Wo/VxJKLXZ37SOnn56Ytb+cc7+6k+2qi+gnGkdYv0gusnMfvicerz6VNRTUDNbDkwooOX5jrnnurkbbuBQufcATP7APCkmU10ztV2dhznnDNr1+TT9vVHgEcASktLO93OM+2SCHIKsNMug9cfbztqqcolpSPFs45/hwxCfT47SkJ180JEJCbyc7PY2UGymZ+b5UE0J/LL9ZmfGghaJy/fnzWZuUvW8/grFeorGMc0cJR/RTUBdc5d0ov31AP14eevmtk7wOnA2nab7jWzkc653WY2EtjX54C91CqJOK7w3DZJqcolpVs6aFHXzQsRkdj50uXj+fIf1rcpw81KS+FLl4/3MKr3+eX6zM8NBO3LOau6mFdS/Eu/R3/yXQmumQ0DDjrnms3sFOA0YGsHmz4N3AZ8M/yzszt28aujpFTkZDpoUdfNCxGR2Gnp5/md5zazq/oY+blZfOny8b7v/9mVZLs+a1/OKfFJv0d/8iwBNbMbgB8Cw4BnzWydc+5y4AJgvpk1AkHgs865g+H3/BT4sXNuLaET22Izux3YDujqWqSFbl6IiHjq+pJRcZlw6vpMRKLNy1FwlwBLOlj/e+D3nbzn062eHwBmRi1AERERkSSj6zMRiTYv5wEVERERERGRJKIEVERERERERGJCCaiIiIiIiIjEhBJQERERERERiQkloCIiIiIiIhIT5pyv5v2NOjOrIjQseGeGAvtjFE4kKe7YiceYIXHjHuOcGxarYPykG+czv4vX72R7ifA5EuEzQPx/Dp3Puicef8/xGDPEZ9zxGDPEZ9xdxdzp+SzpEtCTMbO1zrnSk2/pL4o7duIxZlDc4j+J8rtNhM+RCJ8BEudzSNfi8fccjzFDfMYdjzFDfMbd25hVgisiIiIiIiIxoQRUREREREREYkIJ6Ike8TqAXlLcsROPMYPiFv9JlN9tInyORPgMkDifQ7oWj7/neIwZ4jPueIwZ4jPuXsWsPqAiIiIiIiISE2oBFRERERERkZhQAioiIiIiIiIxkbQJqJldYWabzextM7u3g9czzOw34ddXm1mRB2GeoBtx/5uZvWFm5Wa2wszGeBFnu5i6jLnVdh8xM2dmvhiCujtxm9ms8L/3RjN7PNYxdqQb35FCM/uLmZWFvydXeRFnu5h+bmb7zGxDJ6+bmS0Mf6ZyMzsn1jFK78Xjeau9eD2PtRev57X24vE8J33X3f+HXuvob5qZDTazF8xsS/jnIC9jbM/MRof/z7T83/9ieL3f4840s1fM7PVw3F8Nrx8bvn5/O3w9n+51rO2ZWUr4HPXH8HI8xLzNzNab2TozWxte1/PviHMu6R5ACvAOcAqQDrwOTGi3zZ3Aj8PPbwJ+EydxzwD6hZ9/zuu4uxNzeLts4O/AKqA0Tv6tTwPKgEHh5bw4ifsR4HPh5xOAbT6I+wLgHGBDJ69fBSwDDDgXWO11zHp0+3cbd+et3nyG8Ha+Oo/18nfhu/NaLz+H785zekT/9+6XR0d/04BvA/eGn98LfMvrONvFPBI4J/w8G3gr/H/H73EbMCD8PA1YHb5OWAzcFF7/45bzgZ8ewL8BjwN/DC/HQ8zbgKHt1vX4O5KsLaBTgbedc1udcw3AE8B17ba5DvhF+PnvgJlmZjGMsSMnjds59xfn3NHw4iqgIMYxttedf2uArwHfAupiGVwXuhP3Z4CHnHOHAJxz+2IcY0e6E7cDBoaf5wC7Yhhfh5xzfwcOdrHJdcAvXcgqINfMRsYmOumjeDxvtRev57H24vW81l5cnuekz7r7/9BznfxNa31d+Qvg+ljGdDLOud3OudfCzw8Dm4BR+D9u55w7El5MCz8ccDGh63fwYdxmVgBcDfw0vGz4POYu9Pg7kqwJ6ChgR6vlyvC6DrdxzjUBNcCQmETXue7E3drthFqNvHTSmMPllKOdc8/GMrCT6M6/9enA6Wb2kpmtMrMrYhZd57oT9/3AP5tZJbAU+EJsQuuTnn73xT/i8bzVXryex9qL1/Nae4l6npOuxfvfgeHOud3h53uA4V4G0xULdTsrIdSa6Pu4w6Ws64B9wAuEWsqrw9fv4M/vyv8Ac4BgeHkI/o8ZQsn982b2qpndEV7X4+9IarSiE2+Z2T8DpcCFXsfSFTMLAN8HPuFxKL2RSqhc7SJCLTZ/N7OznHPVXgbVDR8DHnXOfc/MzgN+ZWaTnHPBk71RJJri5bzVXpyfx9qL1/NaezrPiW8555yZ+XIeRDMbAPwe+FfnXG3r4j+/xu2cawYmm1kusAQ4w9uIumZmHwb2OedeNbOLPA6np853zu00szzgBTN7s/WL3f2OJGsL6E5gdKvlgvC6Drcxs1RCJTwHYhJd57oTN2Z2CTAXuNY5Vx+j2DpzspizgUnAX81sG6G6/ad9MIBHd/6tK4GnnXONzrl3CfWXOC1G8XWmO3HfTqifAc65l4FMYGhMouu9bn33xZfi8bzVXryex9qL1/Nae4l6npOuxfvfgb0tXUfCP31X3m5maYSSz8ecc38Ir/Z93C3CN8r+ApxHqKtOS0Ob374r04Frw38vniBUevsD/B0zAM65neGf+wgl+1PpxXckWRPQNcBp4dGm0gkNMvR0u22eBm4LP/8o8GcX7l3roZPGbWYlwMOELuL8cJLoMmbnXI1zbqhzrsg5V0So/9e1zrm13oR7XHe+I08SaiXAzIYSKl3bGsMYO9KduCuAmQBmdiahC7OqmEbZc08DH7eQc4GaVuUe4m/xeN5qL17PY+3F63mtvUQ9z0nXuvN797PW15W3AU95GMsJwn0QfwZscs59v9VLfo97WLjlEzPLAi4l1H/1L4Su38FncTvnvuycKwj/vbiJUI5xCz6OGcDM+ptZdstz4DJgA735jvRlJKR4fhAaVfMtQnXic8Pr5hO6aIDQH6vfAm8DrwCneB1zN+NeDuwF1oUfT/s95nbb/hWfjB7ZjX9rI1R29wawnvDIZV4/uhH3BOAlQiMIrgMu80HMvwZ2A42EWmBuBz4LfLbVv/VD4c+03i/fET26/fuNu/NWTz9Du219cx7rxe/Cl+e1XnwO353n9IjO792Pj07+pg0BVgBbwue8wV7H2S7m8wn17ytvdS6+Kg7iLiY0cnc5oWRoXnj9KYSu398mdD2f4XWsncR/Ee+PguvrmMPxvR5+bGx17u3xd8TCbxQRERERERGJqmQtwRUREREREZEYUwIqIiIiIiIiMaEEVERERERERGJCCaiIiIiIiIjEhBJQERERERERiQkloCIiIiIi0mNm9o8ebn+Rmf0xWvFIfFACKiIiEgcsRH+3RcQ3nHMf9DoGiT/6QyYJxcymmFm5mWWaWX8z22hmk7yOS0SkN8ysyMw2m9kvCU2yPtrrmEREWpjZkfDPi8zsr2b2OzN708weMzMLv3ZFeN1rwD+1em9/M/u5mb1iZmVmdl14/Q/MbF74+eVm9nfdfEss5pzzOgaRiDKzBUAmkAVUOue+4XFIIiK9YmZFwFbgg865VR6HIyLShpkdcc4NMLOLgKeAicAu4CXgS8BaYAtwMfA28Bugn3Puw2b238AbzrlFZpYLvAKUAA5YA9wF/Bi4yjn3Tiw/l0RXqtcBiETBfEInrjpgtsexiIj01XYlnyISB15xzlUCmNk6oAg4ArzrnNsSXr8IuCO8/WXAtWb2H+HlTKDQObfJzD4D/B24W8ln4lECKoloCDAASCN0MnvP23BERPpE5zARiQf1rZ43c/I8w4CPOOc2d/DaWcABID9CsYmPqJ5aEtHDwH8BjwHf8jgWERERkWT1JlBkZqeGlz/W6rXngC+06itaEv45Bvh3QuW4V5rZtBjGKzGgBFQSipl9HGh0zj0OfBOYYmYXexyWiIiISNJxztURKrl9NjwI0b5WL3+NULVauZltBL4WTkZ/BvyHc24XcDvwUzPLjHHoEkUahEhERERERERiQi2gIiIiIiIiEhNKQEVERERERCQmlICKiIiIiIhITCgBFRERERERkZhQAioiIiIiIiIxoQRUREREREREYkIJqIiIiIiIiMTE/wcOgwiATOGZwwAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 936x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "rhs_c_e = model.rhs[c_e].evaluate(0, y0)\n",
    "rhs_c_s = model.rhs[c_s].evaluate(0, y0)\n",
    "rhs = model.concatenated_rhs.evaluate(0, y0)\n",
    "\n",
    "fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(13,4))\n",
    "ax1.plot(x_fine, -10*np.sin(10*x_fine) - 5, x, rhs_c_e, \"o\")\n",
    "ax1.set_xlabel(\"x\")\n",
    "ax1.set_ylabel(\"rhs_c_e\")\n",
    "ax1.legend([\"1+0.1*sin(10*x)\", \"c_e_0\"], loc=\"best\")\n",
    "\n",
    "ax2.plot(r, rhs_c_s, \"o\")\n",
    "ax2.set_xlabel(\"r\")\n",
    "ax2.set_ylabel(\"rhs_c_s\")\n",
    "\n",
    "ax3.plot(rhs,\"*\")\n",
    "ax3.set_xlabel(\"index\")\n",
    "ax3.set_ylabel(\"rhs\")\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `model.concatenated_rhs` is then passed to the solver to solve the model, with initial conditions `model.concatenated_initial_conditions`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Upwinding and downwinding"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If a system is advection-dominated (Peclet number greater than around 40), then it is important to use upwinding (if velocity is positive) or downwinding (if velocity is negative) to obtain accurate results. To see this, consider the following model (without upwinding)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "981ca5da060e40f59a45c5830b64afaf",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='t', step=1.0), Output()), _dom_classes=('widget-inte…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model = pybamm.BaseModel()\n",
    "model.length_scales = {\n",
    "    \"negative electrode\": pybamm.Scalar(1), \n",
    "    \"separator\": pybamm.Scalar(1), \n",
    "    \"positive electrode\": pybamm.Scalar(1)\n",
    "}\n",
    "\n",
    "# Define concentration and velocity\n",
    "c = pybamm.Variable(\"c\", domain=[\"negative electrode\", \"separator\", \"positive electrode\"])\n",
    "v = pybamm.PrimaryBroadcastToEdges(1, [\"negative electrode\", \"separator\", \"positive electrode\"])\n",
    "model.rhs = {c: -pybamm.div(c * v) + 1}\n",
    "model.initial_conditions = {c: 0}\n",
    "model.boundary_conditions = {c: {\"left\": (0, \"Dirichlet\")}}\n",
    "model.variables = {\"c\": c}\n",
    "\n",
    "def solve_and_plot(model):\n",
    "    model_disc = disc.process_model(model, inplace=False)\n",
    "\n",
    "    t_eval = [0,100]\n",
    "    solution = pybamm.CasadiSolver().solve(model_disc, t_eval)\n",
    "\n",
    "    # plot\n",
    "    plot = pybamm.QuickPlot(solution,[\"c\"],spatial_unit=\"m\")\n",
    "    plot.dynamic_plot()\n",
    "    \n",
    "solve_and_plot(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The concentration grows indefinitely, which is clearly an incorrect solution. Instead, we can use upwinding:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f83ad987884941ef907dae01b011ef17",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='t', step=1.0), Output()), _dom_classes=('widget-inte…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model.rhs = {c: -pybamm.div(pybamm.upwind(c) * v) + 1}    \n",
    "solve_and_plot(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This gives the expected linear steady state from 0 to 1. Similarly, if the velocity is negative, downwinding gives accurate results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "2bcaabdc61674c02b24ff1af8cf8c8a4",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='t', step=1.0), Output()), _dom_classes=('widget-inte…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model.rhs = {c: -pybamm.div(pybamm.downwind(c) * (-v)) + 1}   \n",
    "model.boundary_conditions = {c: {\"right\": (0, \"Dirichlet\")}}\n",
    "solve_and_plot(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More advanced concepts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since this notebook is only an introduction to the discretisation, we have not covered everything. More advanced concepts, such as the ones below, can be explored by looking into the [API docs](https://pybamm.readthedocs.io/en/latest/source/spatial_methods/finite_volume.html).\n",
    "\n",
    "- Gradient and divergence of microscale variables in the P2D model\n",
    "- Indefinite integral\n",
    "\n",
    "If you would like detailed examples of these operations, please [create an issue](https://github.com/pybamm-team/PyBaMM/blob/develop/CONTRIBUTING.md#a-before-you-begin) and we will be happy to help."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References\n",
    "\n",
    "The relevant papers for this notebook are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] Joel A. E. Andersson, Joris Gillis, Greg Horn, James B. Rawlings, and Moritz Diehl. CasADi – A software framework for nonlinear optimization and optimal control. Mathematical Programming Computation, 11(1):1–36, 2019. doi:10.1007/s12532-018-0139-4.\n",
      "[2] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, and others. Array programming with NumPy. Nature, 585(7825):357–362, 2020. doi:10.1038/s41586-020-2649-2.\n",
      "[3] Valentin Sulzer, Scott G. Marquis, Robert Timms, Martin Robinson, and S. Jon Chapman. Python Battery Mathematical Modelling (PyBaMM). ECSarXiv. February, 2020. doi:10.1149/osf.io/67ckj.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "pybamm.print_citations()"
   ]
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